cohomology of algebraic spaces contents

COHOMOLOGY OF ALGEBRAIC SPACES

071T

Contents

1. Introduction 12. Conventions 23. Higher direct images 24. Finite morphisms 45. Colimits and cohomology 56. The alternating Čech complex 67. Higher vanishing for quasi-coherent sheaves 118. Vanishing for higher direct images 129. Cohomology with support in a closed subspace 1310. Vanishing above the dimension 1511. Cohomology and base change, I 1612. Coherent modules on locally Noetherian algebraic spaces 1813. Coherent sheaves on Noetherian spaces 2014. Devissage of coherent sheaves 2115. Limits of coherent modules 2516. Vanishing of cohomology 2717. Finite morphisms and affines 3018. A weak version of Chow’s lemma 3119. Noetherian valuative criterion 3320. Higher direct images of coherent sheaves 3521. Ample invertible sheaves and cohomology 3822. The theorem on formal functions 4023. Applications of the theorem on formal functions 4424. Other chapters 45References 46

1. Introduction

071U In this chapter we write about cohomology of algebraic spaces. Although we provesome results on cohomology of abelian sheaves, we focus mainly on cohomologyof quasi-coherent sheaves, i.e., we prove analogues of the results in the chapter“Cohomology of Schemes”. Some of the results in this chapter can be found in[Knu71].An important missing ingredient in this chapter is the induction principle, i.e., theanalogue for quasi-compact and quasi-separated algebraic spaces of Cohomology ofSchemes, Lemma 4.1. This is formulated precisely and proved in detail in Derived

This is a chapter of the Stacks Project, version 86f9bf73, compiled on May 16, 2022.1

COHOMOLOGY OF ALGEBRAIC SPACES 2

Categories of Spaces, Section 9. Instead of the induction principle, in this chapterwe use the alternating Čech complex, see Section 6. It is designed to prove vanishingstatements such as Proposition 7.2, but in some cases the induction principle is amore powerful and perhaps more “standard” tool. We encourage the reader to takea look at the induction principle after reading some of the material in this section.

2. Conventions

071V The standing assumption is that all schemes are contained in a big fppf site Schfppf .And all rings A considered have the property that Spec(A) is (isomorphic) to anobject of this big site.Let S be a scheme and let X be an algebraic space over S. In this chapter and thefollowing we will write X ×S X for the product of X with itself (in the category ofalgebraic spaces over S), instead of X ×X.

3. Higher direct images

071Y Let S be a scheme. Let X be a representable algebraic space over S. Let F be aquasi-coherent module on X (see Properties of Spaces, Section 29). By Descent,Proposition 9.3 the cohomology groups Hi(X,F) agree with the usual cohomologygroup computed in the Zariski topology of the corresponding quasi-coherent moduleon the scheme representing X.More generally, let f : X → Y be a quasi-compact and quasi-separated morphismof representable algebraic spaces X and Y . Let F be a quasi-coherent module on X.By Descent, Lemma 9.5 the sheaf Rif∗F agrees with the usual higher direct imagecomputed for the Zariski topology of the quasi-coherent module on the schemerepresenting X mapping to the scheme representing Y .More generally still, suppose f : X → Y is a representable, quasi-compact, andquasi-separated morphism of algebraic spaces over S. Let V be a scheme and letV → Y be an étale surjective morphism. Let U = V ×Y X and let f ′ : U → V bethe base change of f . Then for any quasi-coherent OX -module F we have(3.0.1)071Z Rif ′∗(F|U ) = (Rif∗F)|V ,see Properties of Spaces, Lemma 26.2. And because f ′ : U → V is a quasi-compact and quasi-separated morphism of schemes, by the remark of the precedingparagraph we may compute Rif ′∗(F|U ) by thinking of F|U as a quasi-coherent sheafon the scheme U , and f ′ as a morphism of schemes. We will frequently use thiswithout further mention.Next, we prove that higher direct images of quasi-coherent sheaves are quasi-coherent for any quasi-compact and quasi-separated morphism of algebraic spaces.In the proof we use a trick; a “better” proof would use a relative Čech complex, asdiscussed in Sheaves on Stacks, Sections 18 and 19 ff.

Lemma 3.1.0720 Let S be a scheme. Let f : X → Y be a morphism of algebraicspaces over S. If f is quasi-compact and quasi-separated, then Rif∗ transformsquasi-coherent OX-modules into quasi-coherent OY -modules.

Proof. Let V → Y be an étale morphism where V is an affine scheme. Set U =V ×Y X and denote f ′ : U → V the induced morphism. Let F be a quasi-coherent OX -module. By Properties of Spaces, Lemma 26.2 we have Rif ′∗(F|U ) =

https://stacks.math.columbia.edu/tag/0720


(Rif∗F)|V . Since the property of being a quasi-coherent module is local in theétale topology on Y (see Properties of Spaces, Lemma 29.6) we may replace Y byV , i.e., we may assume Y is an affine scheme.

Assume Y is affine. Since f is quasi-compact we see that X is quasi-compact. Thuswe may choose an affine scheme U and a surjective étale morphism g : U → X, seeProperties of Spaces, Lemma 6.3. Picture

Ug//

f◦g

X

f

��Y

The morphism g : U → X is representable, separated and quasi-compact becauseX is quasi-separated. Hence the lemma holds for g (by the discussion above thelemma). It also holds for f ◦ g : U → Y (as this is a morphism of affine schemes).

In the situation described in the previous paragraph we will show by induction onn that IHn: for any quasi-coherent sheaf F on X the sheaves RifF are quasi-coherent for i ≤ n. The case n = 0 follows from Morphisms of Spaces, Lemma 11.2.Assume IHn. In the rest of the proof we show that IHn+1 holds.

Let H be a quasi-coherent OU -module. Consider the Leray spectral sequence

Ep,q2 = Rpf∗Rqg∗H ⇒ Rp+q(f ◦ g)∗H

Cohomology on Sites, Lemma 14.7. As Rqg∗H is quasi-coherent by IHn all thesheaves Rpf∗Rqg∗H are quasi-coherent for p ≤ n. The sheaves Rp+q(f ◦ g)∗H areall quasi-coherent (in fact zero for p + q > 0 but we do not need this). Lookingin degrees ≤ n+ 1 the only module which we do not yet know is quasi-coherent isEn+1,0

2 = Rn+1f∗g∗H. Moreover, the differentials dn+1,0r : En+1,0

r → En+1+r,1−rr

are zero as the target is zero. Using that QCoh(OX) is a weak Serre subcategory ofMod(OX) (Properties of Spaces, Lemma 29.7) it follows that Rn+1f∗g∗H is quasi-coherent (details omitted).

Let F be a quasi-coherent OX -module. Set H = g∗F . The adjunction mappingF → g∗g

∗F = g∗H is injective as U → X is surjective étale. Consider the exactsequence

0→ F → g∗H → G → 0where G is the cokernel of the first map and in particular quasi-coherent. Applyingthe long exact cohomology sequence we obtain

Rnf∗g∗H → Rnf∗G → Rn+1f∗F → Rn+1f∗g∗H → Rn+1f∗G

The cokernel of the first arrow is quasi-coherent and we have seen above thatRn+1f∗g∗H is quasi-coherent. Thus Rn+1f∗F has a 2-step filtration where the firststep is quasi-coherent and the second a submodule of a quasi-coherent sheaf. SinceF is an arbitrary quasi-coherent OX -module, this result also holds for G. Thus wecan choose an exact sequence 0 → A → Rn+1f∗G → B with A, B quasi-coherentOY -modules. Then the kernel K of Rn+1f∗g∗H → Rn+1f∗G → B is quasi-coherent,whereupon we obtain a map K → A whose kernel K′ is quasi-coherent too. HenceRn+1f∗F sits in an exact sequence

Rnf∗g∗H → Rnf∗G → Rn+1f∗F → K′ → 0


with all modules quasi-coherent except for possibly Rn+1f∗F . We conclude thatRn+1f∗F is quasi-coherent, i.e., IHn+1 holds as desired. �

Lemma 3.2.08EX Let S be a scheme. Let f : X → Y be a quasi-separated and quasi-compact morphism of algebraic spaces over S. For any quasi-coherent OX-moduleF and any affine object V of Yetale we have

Hq(V ×Y X,F) = H0(V,Rqf∗F)for all q ∈ Z.

Proof. Since formation of Rf∗ commutes with étale localization (Properties ofSpaces, Lemma 26.2) we may replace Y by V and assume Y = V is affine. Considerthe Leray spectral sequence Ep,q2 = Hp(Y,Rqf∗F) converging to Hp+q(X,F), seeCohomology on Sites, Lemma 14.5. By Lemma 3.1 we see that the sheaves Rqf∗Fare quasi-coherent. By Cohomology of Schemes, Lemma 2.2 we see that Ep,q2 = 0when p > 0. Hence the spectral sequence degenerates at E2 and we win. �

4. Finite morphisms

0DK2 Here are some results which hold for all abelian sheaves (in particular also quasi-coherent modules). We warn the reader that these lemmas do not hold for finitemorphisms of schemes and the Zariski topology.

Lemma 4.1.0A4K Let S be a scheme. Let f : X → Y be an integral (for examplefinite) morphism of algebraic spaces. Then f∗ : Ab(Xetale)→ Ab(Yetale) is an exactfunctor and Rpf∗ = 0 for p > 0.

Proof. By Properties of Spaces, Lemma 18.11 we may compute the higher directimages on an étale cover of Y . Hence we may assume Y is a scheme. This impliesthat X is a scheme (Morphisms of Spaces, Lemma 45.3). In this case we may applyÉtale Cohomology, Lemma 43.5. For the finite case the reader may wish to consultthe less technical Étale Cohomology, Proposition 55.2. �

Lemma 4.2.0DK3 Let S be a scheme. Let f : X → Y be a finite morphism of algebraicspaces over S. Let y be a geometric point of Y with lifts x1, . . . , xn in X. Then

(f∗F)y =∏

i=1,...,nFxi

for any sheaf F on Xetale.

Proof. Choose an étale neighbourhood (V, v) of y. Then the stalk (f∗F)y is thestalk of f∗F|V at v. By Properties of Spaces, Lemma 18.11 we may replace Y byV and X by X ×Y V . Then Z → X is a finite morphism of schemes and the resultis Étale Cohomology, Proposition 55.2. �

Lemma 4.3.0DK4 Let S be a scheme. Let π : X → Y be a finite morphism of algebraicspaces over S. Let A be a sheaf of rings on Xetale. Let B be a sheaf of rings onYetale. Let ϕ : B → π∗A be a homomorphism of sheaves of rings so that we obtaina morphism of ringed topoi

f = (π, ϕ) : (Sh(Xetale),A) −→ (Sh(Yetale),B).For a sheaf of A-modules F and a sheaf of B-modules G the canonical map

G ⊗B f∗F −→ f∗(f∗G ⊗A F).is an isomorphism.

https://stacks.math.columbia.edu/tag/08EX

https://stacks.math.columbia.edu/tag/0A4K

https://stacks.math.columbia.edu/tag/0DK3



Proof. The map is the map adjoint to the mapf∗G ⊗A f∗f∗F = f∗(G ⊗B f∗F) −→ f∗G ⊗A F

coming from id : f∗G → f∗G and the adjunction map f∗f∗F → F . To see this mapis an isomorphism, we may check on stalks (Properties of Spaces, Theorem 19.12).Let y be a geometric point of Y and let x1, . . . , xn be the geometric points of Xlying over y. Working out what our maps does on stalks, we see that we have toshow

Gy ⊗By

(⊕i=1,...,n

Fxi

)=⊕

i=1,...,n(Gy ⊗Bx

Axi)⊗Axi

Fxi

which holds true. Here we have used that taking tensor products commutes withtaking stalks, the behaviour of stalks under pullback Properties of Spaces, Lemma19.9, and the behaviour of stalks under pushforward along a closed immersionLemma 4.2. �

We end this section with an insanely general projection formula for finite mor-phisms.

Lemma 4.4.0DK5 With S, X, Y , π, A, B, ϕ, and f as in Lemma 4.3 we have

K ⊗LB Rf∗M = Rf∗(Lf∗K ⊗L

AM)in D(B) for any K ∈ D(B) and M ∈ D(A).

Proof. Since f∗ is exact (Lemma 4.1) the functor Rf∗ is computed by applyingf∗ to any representative complex. Choose a complex K• of B-modules representingK which is K-flat with flat terms, see Cohomology on Sites, Lemma 17.11. Thenf∗K• is K-flat with flat terms, see Cohomology on Sites, Lemma 18.1. Choose anycomplexM• of A-modules representing M . Then we have to show

Tot(K• ⊗B f∗M•) = f∗Tot(f∗K• ⊗AM•)because by our choices these complexes represent the right and left hand side ofthe formula in the lemma. Since f∗ commutes with direct sums (for example bythe description of the stalks in Lemma 4.2), this reduces to the equalities

Kn ⊗B f∗Mm = f∗(f∗Kn ⊗AMm)which are true by Lemma 4.3. �

5. Colimits and cohomology

073D The following lemma in particular applies to diagrams of quasi-coherent sheaves.

Lemma 5.1.073E Let S be a scheme. Let X be an algebraic space over S. If X isquasi-compact and quasi-separated, then

colimiHp(X,Fi) −→ Hp(X, colimi Fi)

is an isomorphism for every filtered diagram of abelian sheaves on Xetale.

Proof. This follows from Cohomology on Sites, Lemma 16.1. Namely, let B ⊂Ob(Xspaces,etale) be the set of quasi-compact and quasi-separated spaces étale overX. Note that if U ∈ B then, because U is quasi-compact, the collection of finitecoverings {Ui → U} with Ui ∈ B is cofinal in the set of coverings of U inXspaces,etale.By Morphisms of Spaces, Lemma 8.10 the set B satisfies all the assumptions ofCohomology on Sites, Lemma 16.1. Since X ∈ B we win. �


https://stacks.math.columbia.edu/tag/073E


Lemma 5.2.07U6 Let S be a scheme. Let f : X → Y be a quasi-compact and quasi-separated morphism of algebraic spaces over S. Let F = colimFi be a filteredcolimit of abelian sheaves on Xetale. Then for any p ≥ 0 we have

Rpf∗F = colimRpf∗Fi.

Proof. Recall that Rpf∗F is the sheaf on Yspaces,etale associated to V 7→ Hp(V ×YX,F), see Cohomology on Sites, Lemma 7.4 and Properties of Spaces, Lemma 18.7.Recall that the colimit is the sheaf associated to the presheaf colimit. Hence we canapply Lemma 5.1 to Hp(V ×Y X,−) where V is affine to conclude (because whenV is affine, then V ×Y X is quasi-compact and quasi-separated). Strictly speakingthis also uses Properties of Spaces, Lemma 18.5 to see that there exist enough affineobjects. �

The following lemma tells us that finitely presented modules behave as expected inquasi-compact and quasi-separated algebraic spaces.

Lemma 5.3.07U7 Let S be a scheme. Let X be a quasi-compact and quasi-separatedalgebraic space over S. Let I be a directed set and let (Fi, ϕii′) be a system over Iof OX-modules. Let G be an OX-module of finite presentation. Then we have

colimi HomX(G,Fi) = HomX(G, colimi Fi).In particular, HomX(G,−) commutes with filtered colimits in QCoh(OX).

Proof. The displayed equality is a special case of Modules on Sites, Lemma 27.12.In order to apply it, we need to check the hypotheses of Sites, Lemma 17.8 part(4) for the site Xetale. In order to do this, we will check hypotheses (2)(a), (2)(b),(2)(c) of Sites, Remark 17.9. Namely, let B ⊂ Ob(Xetale) be the set of affine objects.Then

(1) Since X is quasi-compact, there exists a U ∈ B such that U → X issurjective (Properties of Spaces, Lemma 6.3), hence h#

U → ∗ is surjective.(2) For U ∈ B every étale covering {Ui → U}i∈I of U can be refined by a finite

étale covering {Uj → U}j=1,...,m with Uj ∈ B (Topologies, Lemma 4.4).(3) For U,U ′ ∈ Ob(Xetale) we have h#

U × h#U ′ = h#

U×XU ′. If U,U ′ ∈ B, then

U ×X U ′ is quasi-compact because X is quasi-separated, see Morphismsof Spaces, Lemma 8.10 for example. Hence we can find a surjective étalemorphism U ′′ → U ×X U ′ with U ′′ ∈ B (Properties of Spaces, Lemma 6.3).In other words, we have morphisms U ′′ → U and U ′′ → U ′ such that themap h#

U ′′ → h#U × h

#u′ is surjective.

For the final statement, observe that the inclusion functor QCoh(OX)→ Mod(OX)commutes with colimits and that finitely presented modules are quasi-coherent. SeeProperties of Spaces, Lemma 29.7. �

6. The alternating Čech complex

0721 Let S be a scheme. Let f : U → X be an étale morphism of algebraic spaces overS. The functor

j : Uspaces,etale −→ Xspaces,etale, V/U 7−→ V/X

induces an equivalence of Uspaces,etale with the localization Xspaces,etale/U , seeProperties of Spaces, Section 27. Hence there exist functors

f! : Ab(Uetale) −→ Ab(Xetale), f! : Mod(OU ) −→ Mod(OX),

https://stacks.math.columbia.edu/tag/07U6



which are left adjoint tof−1 : Ab(Xetale) −→ Ab(Uetale), f∗ : Mod(OX) −→ Mod(OU )

see Modules on Sites, Section 19. Warning: This functor, a priori, has nothing todo with cohomology with compact supports! We dubbed this functor “extensionby zero” in the reference above. Note that the two versions of f! agree as f∗ = f−1

for sheaves of OX -modules.As we are going to use this construction below let us recall some of its properties.Given an abelian sheaf G on Uetale the sheaf f! is the sheafification of the presheaf

V/X 7−→ f!G(V ) =⊕

ϕ∈MorX(V,U)G(V ϕ−→ U),

see Modules on Sites, Lemma 19.2. Moreover, if G is an OU -module, then f!G is thesheafification of the exact same presheaf of abelian groups which is endowed withan OX -module structure in an obvious way (see loc. cit.). Let x : Spec(k)→ X bea geometric point. Then there is a canonical identification

(f!G)x =⊕

uGu

where the sum is over all u : Spec(k) → U such that f ◦ u = x, see Modules onSites, Lemma 38.1 and Properties of Spaces, Lemma 19.13. In the following we aregoing to study the sheaf f!Z. Here Z denotes the constant sheaf on Xetale or Uetale.

Lemma 6.1.0722 Let S be a scheme. Let fi : Ui → X be étale morphisms of algebraicspaces over S. Then there are isomorphisms

f1,!Z⊗Z f2,!Z −→ f12,!Zwhere f12 : U1 ×X U2 → X is the structure morphism and

(f1 q f2)!Z −→ f1,!Z⊕ f2,!Z

Proof. Once we have defined the map it will be an isomorphism by our descriptionof stalks above. To define the map it suffices to work on the level of presheaves.Thus we have to define a map(⊕

ϕ1∈MorX(V,U1)Z)⊗Z

(⊕ϕ2∈MorX(V,U2)

Z)−→

⊕ϕ∈MorX(V,U1×XU2)

Z

We map the element 1ϕ1 ⊗ 1ϕ2 to the element 1ϕ1×ϕ2 with obvious notation. Weomit the proof of the second equality. �

Another important feature is the trace mapTrf : f!Z −→ Z.

The trace map is adjoint to the map Z→ f−1Z (which is an isomorphism). If x isabove, then Trf on stalks at x is the map

(Trf )x : (f!Z)x =⊕

uZ −→ Z = Zx

which sums the given integers. This is true because it is adjoint to the map 1 : Z→f−1Z. In particular, if f is surjective as well as étale then Trf is surjective.Assume that f : U → X is a surjective étale morphism of algebraic spaces. Considerthe Koszul complex associated to the trace map we discussed above

. . .→ ∧3f!Z→ ∧2f!Z→ f!Z→ Z→ 0



Here the exterior powers are over the sheaf of rings Z. The maps are defined bythe rule

e1 ∧ . . . ∧ en 7−→∑

i=1,...,n(−1)i+1Trf (ei)e1 ∧ . . . ∧ ei ∧ . . . ∧ en

where e1, . . . , en are local sections of f!Z. Let x be a geometric point of X and setMx = (f!Z)x =

⊕u Z. Then the stalk of the complex above at x is the complex

. . .→ ∧3Mx → ∧2Mx →Mx → Z→ 0which is exact because Mx → Z is surjective, see More on Algebra, Lemma 28.5.Hence if we let K• = K•(f) be the complex with Ki = ∧i+1f!Z, then we obtain aquasi-isomorphism(6.1.1)0723 K• −→ Z[0]

We use the complex K• to define what we call the alternating Čech complex asso-ciated to f : U → X.

Definition 6.2.0724 Let S be a scheme. Let f : U → X be a surjective étale morphismof algebraic spaces over S. Let F be an object of Ab(Xetale). The alternating Čechcomplex1 C•alt(f,F) associated to F and f is the complex

Hom(K0,F)→ Hom(K1,F)→ Hom(K2,F)→ . . .

with Hom groups computed in Ab(Xetale).

The reader may verify that if U =∐Ui and f |Ui : Ui → X is the open immersion

of a subspace, then C•alt(f,F) agrees with the complex introduced in Cohomology,Section 23 for the Zariski covering X =

⋃Ui and the restriction of F to the Zariski

site of X. What is more important however, is to relate the cohomology of thealternating Čech complex to the cohomology.

Lemma 6.3.0725 Let S be a scheme. Let f : U → X be a surjective étale morphism ofalgebraic spaces over S. Let F be an object of Ab(Xetale). There exists a canonicalmap

C•alt(f,F) −→ RΓ(X,F)in D(Ab). Moreover, there is a spectral sequence with E1-page

Ep,q1 = ExtqAb(Xetale)(Kp,F)

converging to Hp+q(X,F) where Kp = ∧p+1f!Z.

Proof. Recall that we have the quasi-isomorphism K• → Z[0], see (6.1.1). Choosean injective resolution F → I• inAb(Xetale). Consider the double complex Hom(K•, I•)with terms Hom(Kp, Iq). The differential dp,q1 : Ap,q → Ap+1,q is the one comingfrom the differential Kp+1 → Kp and the differential dp,q2 : Ap,q → Ap,q+1 is the onecoming from the differential Iq → Iq+1. Denote Tot(Hom(K•, I•)) the associatedtotal complex, see Homology, Section 18. We will use the two spectral sequences(′Er, ′dr) and (′′Er, ′′dr) associated to this double complex, see Homology, Section25.Because K• is a resolution of Z we see that the complexes

Hom(K•, Iq) : Hom(K0, Iq)→ Hom(K1, Iq)→ Hom(K2, Iq)→ . . .

1This may be nonstandard notation




are acyclic in positive degrees and have H0 equal to Γ(X, Iq). Hence by Homology,Lemma 25.4 the natural map

I•(X) −→ Tot(Hom(K•, I•))is a quasi-isomorphism of complexes of abelian groups. In particular we concludethat Hn(Tot(Hom(K•, I•))) = Hn(X,F).The map C•alt(f,F) → RΓ(X,F) of the lemma is the composition of C•alt(f,F) →Tot(Hom(K•, I•)) with the inverse of the displayed quasi-isomorphism.Finally, consider the spectral sequence (′Er, ′dr). We haveEp,q1 = qth cohomology of Hom(Kp, I0)→ Hom(Kp, I1)→ Hom(Kp, I2)→ . . .

This proves the lemma. �

It follows from the lemma that it is important to understand the ext groupsExtAb(Xetale)(Kp,F), i.e., the right derived functors of F 7→ Hom(Kp,F).

Lemma 6.4.0726 Let S be a scheme. Let f : U → X be a surjective, étale, andseparated morphism of algebraic spaces over S. For p ≥ 0 set

Wp = U ×X . . .×X U \ all diagonalswhere the fibre product has p+ 1 factors. There is a free action of Sp+1 on Wp overX and

Hom(Kp,F) = Sp+1-anti-invariant elements of F(Wp)functorially in F where Kp = ∧p+1f!Z.

Proof. Because U → X is separated the diagonal U → U ×X U is a closed immer-sion. Since U → X is étale the diagonal U → U ×X U is an open immersion, seeMorphisms of Spaces, Lemmas 39.10 and 38.9. Hence Wp is an open and closedsubspace of Up+1 = U ×X . . . ×X U . The action of Sp+1 on Wp is free as we’vethrown out the fixed points of the action. By Lemma 6.1 we see that

(f!Z)⊗p+1 = fp+1! Z = (Wp → X)!Z⊕Rest

where fp+1 : Up+1 → X is the structure morphism. Looking at stalks over ageometric point x of X we see that(⊕

u7→xZ)⊗p+1

−→ (Wp → X)!Zxis the quotient whose kernel is generated by all tensors 1u0⊗ . . .⊗1up

where ui = ujfor some i 6= j. Thus the quotient map

(f!Z)⊗p+1 −→ ∧p+1f!Zfactors through (Wp → X)!Z, i.e., we get

(f!Z)⊗p+1 −→ (Wp → X)!Z −→ ∧p+1f!ZThis already proves that Hom(Kp,F) is (functorially) a subgroup of

Hom((Wp → X)!Z,F) = F(Wp)To identify it with the Sp+1-anti-invariants we have to prove that the surjection(Wp → X)!Z → ∧p+1f!Z is the maximal Sp+1-anti-invariant quotient. In otherwords, we have to show that ∧p+1f!Z is the quotient of (Wp → X)!Z by thesubsheaf generated by the local sections s − sign(σ)σ(s) where s is a local sectionof (Wp → X)!Z. This can be checked on the stalks, where it is clear. �



Lemma 6.5.0727 Let S be a scheme. Let W be an algebraic space over S. Let G bea finite group acting freely on W . Let U = W/G, see Properties of Spaces, Lemma34.1. Let χ : G→ {+1,−1} be a character. Then there exists a rank 1 locally freesheaf of Z-modules Z(χ) on Uetale such that for every abelian sheaf F on Uetale wehave

H0(W,F|W )χ = H0(U,F ⊗Z Z(χ))

Proof. The quotient morphism q : W → U is a G-torsor, i.e., there exists asurjective étale morphism U ′ → U such that W ×U U ′ =

∐g∈G U

′ as spaces withG-action over U ′. (Namely, U ′ = W works.) Hence q∗Z is a finite locally freeZ-module with an action of G. For any geometric point u of U , then we get G-equivariant isomorphisms

(q∗Z)u =⊕

w 7→uZ =

⊕g∈G

Z = Z[G]

where the second = uses a geometric point w0 lying over u and maps the summandcorresponding to g ∈ G to the summand corresponding to g(w0). We have

H0(W,F|W ) = H0(U,F ⊗Z q∗Z)because q∗F|W = F ⊗Z q∗Z as one can check by restricting to U ′. Let

Z(χ) = (q∗Z)χ ⊂ q∗Zbe the subsheaf of sections that transform according to χ. For any geometric pointu of U we have

Z(χ)u = Z ·∑

gχ(g)g ⊂ Z[G] = (q∗Z)u

It follows that Z(χ) is locally free of rank 1 (more precisely, this should be checkedafter restricting to U ′). Note that for any Z-module M the χ-semi-invariants ofM [G] are the elements of the form m ·

∑g χ(g)g. Thus we see that for any abelian

sheaf F on U we have(F ⊗Z q∗Z)χ = F ⊗Z Z(χ)

because we have equality at all stalks. The result of the lemma follows by takingglobal sections. �

Now we can put everything together and obtain the following pleasing result.

Lemma 6.6.0728 Let S be a scheme. Let f : U → X be a surjective, étale, andseparated morphism of algebraic spaces over S. For p ≥ 0 set

Wp = U ×X . . .×X U \ all diagonals(with p + 1 factors) as in Lemma 6.4. Let χp : Sp+1 → {+1,−1} be the signcharacter. Let Up = Wp/Sp+1 and Z(χp) be as in Lemma 6.5. Then the spectralsequence of Lemma 6.3 has E1-page

Ep,q1 = Hq(Up,F|Up⊗Z Z(χp))

and converges to Hp+q(X,F).

Proof. Note that since the action of Sp+1 onWp is overX we do obtain a morphismUp → X. Since Wp → X is étale and since Wp → Up is surjective étale, it followsthat also Up → X is étale, see Morphisms of Spaces, Lemma 39.2. Therefore aninjective object of Ab(Xetale) restricts to an injective object of Ab(Up,etale), seeCohomology on Sites, Lemma 7.1. Moreover, the functor G 7→ G ⊗Z Z(χp)) isan auto-equivalence of Ab(Up), whence transforms injective objects into injective




objects and is exact (because Z(χp) is an invertible Z-module). Thus given aninjective resolution F → I• in Ab(Xetale) the complex

Γ(Up, I0|Up⊗Z Z(χp))→ Γ(Up, I1|Up

⊗Z Z(χp))→ Γ(Up, I2|Up⊗Z Z(χp))→ . . .

computes H∗(Up,F|Up⊗Z Z(χp)). On the other hand, by Lemma 6.5 it is equal to

the complex of Sp+1-anti-invariants in

Γ(Wp, I0)→ Γ(Wp, I1)→ Γ(Wp, I2)→ . . .

which by Lemma 6.4 is equal to the complex

Hom(Kp, I0)→ Hom(Kp, I1)→ Hom(Kp, I2)→ . . .

which computes Ext∗Ab(Xetale)(Kp,F). Putting everything together we win. �

7. Higher vanishing for quasi-coherent sheaves

0729 In this section we show that given a quasi-compact and quasi-separated algebraicspace X there exists an integer n = n(X) such that the cohomology of any quasi-coherent sheaf on X vanishes beyond degree n.

Lemma 7.1.072A With S, W , G, U , χ as in Lemma 6.5. If F is a quasi-coherentOU -module, then so is F ⊗Z Z(χ).

Proof. The OU -module structure is clear. To check that F ⊗Z Z(χ) is quasi-coherent it suffices to check étale locally. Hence the lemma follows as Z(χ) is finitelocally free as a Z-module. �

The following proposition is interesting even if X is a scheme. It is the naturalgeneralization of Cohomology of Schemes, Lemma 4.2. Before we state it, observethat given an étale morphism f : U → X from an affine scheme towards a quasi-separated algebraic space X the fibres of f are universally bounded, in particularthere exists an integer d such that the fibres of |U | → |X| all have size at most d;this is the implication (η)⇒ (δ) of Decent Spaces, Lemma 5.1.

Proposition 7.2.072B Let S be a scheme. Let X be an algebraic space over S. AssumeX is quasi-compact and separated. Let U be an affine scheme, and let f : U → Xbe a surjective étale morphism. Let d be an upper bound for the size of the fibres of|U | → |X|. Then for any quasi-coherent OX-module F we have Hq(X,F) = 0 forq ≥ d.

Proof. We will use the spectral sequence of Lemma 6.6. The lemma applies sincef is separated as U is separated, see Morphisms of Spaces, Lemma 4.10. Since X isseparated the scheme U×X . . .×XU is a closed subscheme of U×Spec(Z). . .×Spec(Z)Uhence is affine. Thus Wp is affine. Hence Up = Wp/Sp+1 is an affine scheme byGroupoids, Proposition 23.9. The discussion in Section 3 shows that cohomology ofquasi-coherent sheaves on Wp (as an algebraic space) agrees with the cohomologyof the corresponding quasi-coherent sheaf on the underlying affine scheme, hencevanishes in positive degrees by Cohomology of Schemes, Lemma 2.2. By Lemma7.1 the sheaves F|Up ⊗Z Z(χp) are quasi-coherent. Hence Hq(Wp,F|Up

⊗Z Z(χp))is zero when q > 0. By our definition of the integer d we see that Wp = ∅ forp ≥ d. Hence also H0(Wp,F|Up ⊗Z Z(χp)) is zero when p ≥ d. This proves theproposition. �

https://stacks.math.columbia.edu/tag/072A

https://stacks.math.columbia.edu/tag/072B


In the following lemma we establish that a quasi-compact and quasi-separated al-gebraic space has finite cohomological dimension for quasi-coherent modules. Weare explicit about the bound only because we will use it later to prove a similarresult for higher direct images.

Lemma 7.3.072C Let S be a scheme. Let X be an algebraic space over S. Assume Xis quasi-compact and quasi-separated. Then we can choose

(1) an affine scheme U ,(2) a surjective étale morphism f : U → X,(3) an integer d bounding the degrees of the fibres of U → X,(4) for every p = 0, 1, . . . , d a surjective étale morphism Vp → Up from an affine

scheme Vp where Up is as in Lemma 6.6, and(5) an integer dp bounding the degree of the fibres of Vp → Up.

Moreover, whenever we have (1) – (5), then for any quasi-coherent OX-module Fwe have Hq(X,F) = 0 for q ≥ max(dp + p).

Proof. Since X is quasi-compact we can find a surjective étale morphism U → Xwith U affine, see Properties of Spaces, Lemma 6.3. By Decent Spaces, Lemma 5.1the fibres of f are universally bounded, hence we can find d. We have Up = Wp/Sp+1and Wp ⊂ U ×X . . . ×X U is open and closed. Since X is quasi-separated theschemes Wp are quasi-compact, hence Up is quasi-compact. Since U is separated,the schemes Wp are separated, hence Up is separated by (the absolute versionof) Spaces, Lemma 14.5. By Properties of Spaces, Lemma 6.3 we can find themorphisms Vp →Wp. By Decent Spaces, Lemma 5.1 we can find the integers dp.

At this point the proof uses the spectral sequence

Ep,q1 = Hq(Up,F|Up⊗Z Z(χp))⇒ Hp+q(X,F)

see Lemma 6.6. By definition of the integer d we see that Up = 0 for p ≥ d. ByProposition 7.2 and Lemma 7.1 we see thatHq(Up,F|Up⊗ZZ(χp)) is zero for q ≥ dpfor p = 0, . . . , d. Whence the lemma. �

8. Vanishing for higher direct images

073F We apply the results of Section 7 to obtain vanishing of higher direct images ofquasi-coherent sheaves for quasi-compact and quasi-separated morphisms. This isuseful because it allows one to argue by descending induction on the cohomologicaldegree in certain situations.

Lemma 8.1.073G Let S be a scheme. Let f : X → Y be a morphism of algebraicspaces over S. Assume that

(1) f is quasi-compact and quasi-separated, and(2) Y is quasi-compact.

Then there exists an integer n(X → Y ) such that for any algebraic space Y ′, anymorphism Y ′ → Y and any quasi-coherent sheaf F ′ on X ′ = Y ′ ×Y X the higherdirect images Rif ′∗F ′ are zero for i ≥ n(X → Y ).

Proof. Let V → Y be a surjective étale morphism where V is an affine scheme, seeProperties of Spaces, Lemma 6.3. Suppose we prove the result for the base changefV : V ×Y X → V . Then the result holds for f with n(X → Y ) = n(XV → V ).Namely, if Y ′ → Y and F ′ are as in the lemma, then Rif ′∗F ′|V×Y Y ′ is equal to

https://stacks.math.columbia.edu/tag/072C

https://stacks.math.columbia.edu/tag/073G


Rif ′V,∗F ′|X′V where f ′V : X ′V = V ×Y Y ′ ×Y X → V ×Y Y ′ = Y ′V , see Properties ofSpaces, Lemma 26.2. Thus we may assume that Y is an affine scheme.

Moreover, to prove the vanishing for all Y ′ → Y and F ′ it suffices to do so whenY ′ is an affine scheme. In this case, Rif ′∗F ′ is quasi-coherent by Lemma 3.1. Henceit suffices to prove that Hi(X ′,F ′) = 0, because Hi(X ′,F ′) = H0(Y ′, Rif ′∗F ′)by Cohomology on Sites, Lemma 14.6 and the vanishing of higher cohomology ofquasi-coherent sheaves on affine algebraic spaces (Proposition 7.2).

Choose U → X, d, Vp → Up and dp as in Lemma 7.3. For any affine scheme Y ′ andmorphism Y ′ → Y denote X ′ = Y ′ ×Y X, U ′ = Y ′ ×Y U , V ′p = Y ′ ×Y Vp. ThenU ′ → X ′, d′ = d, V ′p → U ′p and d′p = d is a collection of choices as in Lemma 7.3for the algebraic space X ′ (details omitted). Hence we see that Hi(X ′,F ′) = 0 fori ≥ max(p+ dp) and we win. �

Lemma 8.2.073H Let S be a scheme. Let f : X → Y be an affine morphism of algebraicspaces over S. Then Rif∗F = 0 for i > 0 and any quasi-coherent OX-module F .

Proof. Recall that an affine morphism of algebraic spaces is representable. Hencethis follows from (3.0.1) and Cohomology of Schemes, Lemma 2.3. �

Lemma 8.3.0D2U Let S be a scheme. Let f : X → Y be an affine morphism ofalgebraic spaces over S. Let F be a quasi-coherent OX-module. Then Hi(X,F) =Hi(Y, f∗F) for all i ≥ 0.

Proof. Follows from Lemma 8.2 and the Leray spectral sequence. See Cohomologyon Sites, Lemma 14.6. �

9. Cohomology with support in a closed subspace

0A4L This section is the analogue of Cohomology, Sections 21 and 34 and Étale Coho-mology, Section 79 for abelian sheaves on algebraic spaces.

Let S be a scheme. Let X be an algebraic space over S and let Z ⊂ X be a closedsubspace. Let F be an abelian sheaf on Xetale. We let

ΓZ(X,F) = {s ∈ F(X) | Supp(s) ⊂ Z}

be the sections with support in Z (Properties of Spaces, Definition 20.3). This is aleft exact functor which is not exact in general. Hence we obtain a derived functor

RΓZ(X,−) : D(Xetale) −→ D(Ab)

and cohomology groups with support in Z defined by HqZ(X,F) = RqΓZ(X,F).

Let I be an injective abelian sheaf on Xetale. Let U ⊂ X be the open subspacewhich is the complement of Z. Then the restriction map I(X)→ I(U) is surjective(Cohomology on Sites, Lemma 12.6) with kernel ΓZ(X, I). It immediately followsthat for K ∈ D(Xetale) there is a distinguished triangle

RΓZ(X,K)→ RΓ(X,K)→ RΓ(U,K)→ RΓZ(X,K)[1]

in D(Ab). As a consequence we obtain a long exact cohomology sequence

. . .→ HiZ(X,K)→ Hi(X,K)→ Hi(U,K)→ Hi+1

Z (X,K)→ . . .

for any K in D(Xetale).

https://stacks.math.columbia.edu/tag/073H

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For an abelian sheaf F on Xetale we can consider the subsheaf of sections withsupport in Z, denoted HZ(F), defined by the rule

HZ(F)(U) = {s ∈ F(U) | Supp(s) ⊂ U ×X Z}

Here we use the support of a section from Properties of Spaces, Definition 20.3.Using the equivalence of Morphisms of Spaces, Lemma 13.5 we may view HZ(F)as an abelian sheaf on Zetale. Thus we obtain a functor

Ab(Xetale) −→ Ab(Zetale), F 7−→ HZ(F)

which is left exact, but in general not exact.

Lemma 9.1.0A4M Let S be a scheme. Let i : Z → X be a closed immersion of algebraicspaces over S. Let I be an injective abelian sheaf on Xetale. Then HZ(I) is aninjective abelian sheaf on Zetale.

Proof. Observe that for any abelian sheaf G on Zetale we have

HomZ(G,HZ(F)) = HomX(i∗G,F)

because after all any section of i∗G has support in Z. Since i∗ is exact (Lemma 4.1)and as I is injective on Xetale we conclude that HZ(I) is injective on Zetale. �

DenoteRHZ : D(Xetale) −→ D(Zetale)

the derived functor. We set HqZ(F) = RqHZ(F) so that H0Z(F) = HZ(F). By the

lemma above we have a Grothendieck spectral sequence

Ep,q2 = Hp(Z,HqZ(F))⇒ Hp+qZ (X,F)

Lemma 9.2.0A4N Let S be a scheme. Let i : Z → X be a closed immersion of algebraicspaces over S. Let G be an injective abelian sheaf on Zetale. Then HpZ(i∗G) = 0 forp > 0.

Proof. This is true because the functor i∗ is exact (Lemma 4.1) and transforms in-jective abelian sheaves into injective abelian sheaves (Cohomology on Sites, Lemma14.2). �

Lemma 9.3.0A4P Let S be a scheme. Let f : X → Y be an étale morphism ofalgebraic spaces over S. Let Z ⊂ Y be a closed subspace such that f−1(Z) → Z isan isomorphism of algebraic spaces. Let F be an abelian sheaf on X. Then

HqZ(F) = Hqf−1(Z)(f−1F)

as abelian sheaves on Z = f−1(Z) and we have HqZ(Y,F) = Hq

f−1(Z)(X, f−1F).

Proof. Because f is étale an injective resolution of F pulls back to an injectiveresolution of f−1F . Hence it suffices to check the equality for HZ(−) which followsfrom the definitions. The proof for cohomology with supports is the same. Somedetails omitted. �

Let S be a scheme and let X be an algebraic space over S. Let T ⊂ |X| be a closedsubset. We denote DT (Xetale) the strictly full saturated triangulated subcategoryof D(Xetale) consisting of objects whose cohomology sheaves are supported on T .

https://stacks.math.columbia.edu/tag/0A4M

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https://stacks.math.columbia.edu/tag/0A4P


Lemma 9.4.0AEI Let S be a scheme. Let i : Z → X be a closed immersion of algebraicspaces over S. The map Ri∗ = i∗ : D(Zetale) → D(Xetale) induces an equivalenceD(Zetale)→ D|Z|(Xetale) with quasi-inverse

i−1|DZ(Xetale) = RHZ |D|Z|(Xetale)

Proof. Recall that i−1 and i∗ is an adjoint pair of exact functors such that i−1i∗ isisomorphic to the identify functor on abelian sheaves. See Properties of Spaces,Lemma 19.9 and Morphisms of Spaces, Lemma 13.5. Thus i∗ : D(Zetale) →DZ(Xetale) is fully faithful and i−1 determines a left inverse. On the other hand,suppose that K is an object of DZ(Xetale) and consider the adjunction map K →i∗i−1K. Using exactness of i∗ and i−1 this induces the adjunction maps Hn(K)→

i∗i−1Hn(K) on cohomology sheaves. Since these cohomology sheaves are sup-

ported on Z we see these adjunction maps are isomorphisms and we conclude thatD(Zetale)→ DZ(Xetale) is an equivalence.

To finish the proof we have to show that RHZ(K) = i−1K if K is an object ofDZ(Xetale). To do this we can use that K = i∗i

−1K as we’ve just proved this isthe case. Then we can choose a K-injective representative I• for i−1K. Since i∗ isthe right adjoint to the exact functor i−1, the complex i∗I• is K-injective (DerivedCategories, Lemma 31.9). We see that RHZ(K) is computed by HZ(i∗I•) = I• asdesired. �

10. Vanishing above the dimension

0A4Q Let S be a scheme. Let X be a quasi-compact and quasi-separated algebraic spaceover S. In this case |X| is a spectral space, see Properties of Spaces, Lemma 15.2.Moreover, the dimension of X (as defined in Properties of Spaces, Definition 9.2) isequal to the Krull dimension of |X|, see Decent Spaces, Lemma 12.5. We will showthat for quasi-coherent sheaves on X we have vanishing of cohomology above thedimension. This result is already interesting for quasi-separated algebraic spaces offinite type over a field.

Lemma 10.1.0A4R Let S be a scheme. Let X be a quasi-compact and quasi-separatedalgebraic space over S. Assume dim(X) ≤ d for some integer d. Let F be a quasi-coherent sheaf F on X.

(1) Hq(X,F) = 0 for q > d,(2) Hd(X,F)→ Hd(U,F) is surjective for any quasi-compact open U ⊂ X,(3) Hq

Z(X,F) = 0 for q > d for any closed subspace Z ⊂ X whose complementis quasi-compact.

Proof. By Properties of Spaces, Lemma 22.5 every algebraic space Y étale overX has dimension ≤ d. If Y is quasi-separated, the dimension of Y is equal to theKrull dimension of |Y | by Decent Spaces, Lemma 12.5. Also, if Y is a scheme,then étale cohomology of F over Y , resp. étale cohomology of F with support ina closed subscheme, agrees with usual cohomology of F , resp. usual cohomologywith support in the closed subscheme. See Descent, Proposition 9.3 and ÉtaleCohomology, Lemma 79.5. We will use these facts without further mention.

By Decent Spaces, Lemma 8.6 there exist an integer n and open subspaces

∅ = Un+1 ⊂ Un ⊂ Un−1 ⊂ . . . ⊂ U1 = X

https://stacks.math.columbia.edu/tag/0AEI

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with the following property: setting Tp = Up\Up+1 (with reduced induced subspacestructure) there exists a quasi-compact separated scheme Vp and a surjective étalemorphism fp : Vp → Up such that f−1

p (Tp)→ Tp is an isomorphism.

As Un = Vn is a scheme, our initial remarks imply the cohomology of F over Unvanishes in degrees > d by Cohomology, Proposition 22.4. Suppose we have shown,by induction, that Hq(Up+1,F|Up+1) = 0 for q > d. It suffices to show Hq

Tp(Up,F)

for q > d is zero in order to conclude the vanishing of cohomology of F over Up indegrees > d. However, we have

HqTp

(Up,F) = Hq

f−1p (Tp)(Vp,F)

by Lemma 9.3 and as Vp is a scheme we obtain the desired vanishing from Coho-mology, Proposition 22.4. In this way we conclude that (1) is true.

To prove (2) let U ⊂ X be a quasi-compact open subspace. Consider the open sub-space U ′ = U ∪ Un. Let Z = U ′ \ U . Then g : Un → U ′ is an étale morphism suchthat g−1(Z) → Z is an isomorphism. Hence by Lemma 9.3 we have Hq

Z(U ′,F) =HqZ(Un,F) which vanishes in degree > d because Un is a scheme and we can apply

Cohomology, Proposition 22.4. We conclude that Hd(U ′,F)→ Hd(U,F) is surjec-tive. Assume, by induction, that we have reduced our problem to the case whereU contains Up+1. Then we set U ′ = U ∪ Up, set Z = U ′ \ U , and we argue usingthe morphism fp : Vp → U ′ which is étale and has the property that f−1

p (Z) → Zis an isomorphism. In other words, we again see that

HqZ(U ′,F) = Hq

f−1p (Z)(Vp,F)

and we again see this vanishes in degrees > d. We conclude that Hd(U ′,F) →Hd(U,F) is surjective. Eventually we reach the stage where U1 = X ⊂ U whichfinishes the proof.

A formal argument shows that (2) implies (3). �

11. Cohomology and base change, I

073I Let S be a scheme. Let f : X → Y be a morphism of algebraic spaces over S. LetF be a quasi-coherent sheaf on X. Suppose further that g : Y ′ → Y is a morphismof algebraic spaces over S. Denote X ′ = XY ′ = Y ′×Y X the base change of X anddenote f ′ : X ′ → Y ′ the base change of f . Also write g′ : X ′ → X the projection,and set F ′ = (g′)∗F . Here is a diagram representing the situation:

(11.0.1)073J

F ′ = (g′)∗F X ′g′//

f ′

��

X

f

��

F

Rf ′∗F ′ Y ′g // Y Rf∗F

Here is the simplest case of the base change property we have in mind.

Lemma 11.1.07U8 Let S be a scheme. Let f : X → Y be an affine morphism ofalgebraic spaces over S. Let F be a quasi-coherent OX-module. In this case f∗F ∼=Rf∗F is a quasi-coherent sheaf, and for every diagram (11.0.1) we have

g∗f∗F = f ′∗(g′)∗F .



Proof. By the discussion surrounding (3.0.1) this reduces to the case of an affinemorphism of schemes which is treated in Cohomology of Schemes, Lemma 5.1. �

Lemma 11.2 (Flat base change).073K Let S be a scheme. Consider a cartesian dia-gram of algebraic spaces

X ′

f ′

��

g′// X

f

��Y ′

g // Y

over S. Let F be a quasi-coherent OX-module with pullback F ′ = (g′)∗F . Assumethat g is flat and that f is quasi-compact and quasi-separated. For any i ≥ 0

(1) the base change map of Cohomology on Sites, Lemma 15.1 is an isomor-phism

g∗Rif∗F −→ Rif ′∗F ′,

(2) if Y = Spec(A) and Y ′ = Spec(B), then Hi(X,F)⊗A B = Hi(X ′,F ′).

Proof. The morphism g′ is flat by Morphisms of Spaces, Lemma 30.4. Note thatflatness of g and g′ is equivalent to flatness of the morphisms of small étale ringedsites, see Morphisms of Spaces, Lemma 30.9. Hence we can apply Cohomology onSites, Lemma 15.1 to obtain a base change map

g∗Rpf∗F −→ Rpf ′∗F ′

To prove this map is an isomorphism we can work locally in the étale topology onY ′. Thus we may assume that Y and Y ′ are affine schemes. Say Y = Spec(A) andY ′ = Spec(B). In this case we are really trying to show that the map

Hp(X,F)⊗A B −→ Hp(XB ,FB)

is an isomorphism where XB = Spec(B) ×Spec(A) X and FB is the pullback of Fto XB . In other words, it suffices to prove (2).

Fix A → B a flat ring map and let X be a quasi-compact and quasi-separatedalgebraic space over A. Note that g′ : XB → X is affine as a base change ofSpec(B)→ Spec(A). Hence the higher direct images Ri(g′)∗FB are zero by Lemma8.2. Thus Hp(XB ,FB) = Hp(X, g′∗FB), see Cohomology on Sites, Lemma 14.6.Moreover, we have

g′∗FB = F ⊗A B

where A, B denotes the constant sheaf of rings with value A, B. Namely, it is clearthat there is a map from right to left. For any affine scheme U étale over X wehave

g′∗FB(U) = FB(Spec(B)×Spec(A) U)= Γ(Spec(B)×Spec(A) U, (Spec(B)×Spec(A) U → U)∗F|U )= B ⊗A F(U)

hence the map is an isomorphism. Write B = colimMi as a filtered colimit of finitefree A-modulesMi using Lazard’s theorem, see Algebra, Theorem 81.4. We deduce

https://stacks.math.columbia.edu/tag/073K


thatHp(X, g′∗FB) = Hp(X,F ⊗A B)

= Hp(X, colimi F ⊗AMi)= colimiH

p(X,F ⊗AMi)= colimiH

p(X,F)⊗AMi

= Hp(X,F)⊗A colimiMi

= Hp(X,F)⊗A BThe first equality because g′∗FB = F ⊗A B as seen above. The second because⊗ commutes with colimits. The third equality because cohomology on X com-mutes with colimits (see Lemma 5.1). The fourth equality because Mi is finite free(i.e., because cohomology commutes with finite direct sums). The fifth because ⊗commutes with colimits. The sixth by choice of our system. �

12. Coherent modules on locally Noetherian algebraic spaces

07U9 This section is the analogue of Cohomology of Schemes, Section 9. In Modules onSites, Definition 23.1 we have defined coherent modules on any ringed topos. Weuse this notion to define coherent modules on locally Noetherian algebraic spaces.Although it is possible to work with coherent modules more generally we resist theurge to do so.Definition 12.1.07UA Let S be a scheme. Let X be a locally Noetherian algebraicspace over S. A quasi-coherent module F on X is called coherent if F is a coherentOX -module on the site Xetale in the sense of Modules on Sites, Definition 23.1.Of course this definition is a bit hard to work with. We usually use the characteri-zation given in the lemma below.Lemma 12.2.07UB Let S be a scheme. Let X be a locally Noetherian algebraic spaceover S. Let F be an OX-module. The following are equivalent

(1) F is coherent,(2) F is a quasi-coherent, finite type OX-module,(3) F is a finitely presented OX-module,(4) for any étale morphism ϕ : U → X where U is a scheme the pullback ϕ∗F

is a coherent module on U , and(5) there exists a surjective étale morphism ϕ : U → X where U is a scheme

such that the pullback ϕ∗F is a coherent module on U .In particular OX is coherent, any invertible OX-module is coherent, and more gen-erally any finite locally free OX-module is coherent.Proof. To be sure, if X is a locally Noetherian algebraic space and U → X is anétale morphism, then U is locally Noetherian, see Properties of Spaces, Section 7.The lemma then follows from the points (1) – (5) made in Properties of Spaces,Section 30 and the corresponding result for coherent modules on locally Noetherianschemes, see Cohomology of Schemes, Lemma 9.1. �

Lemma 12.3.07UC Let S be a scheme. Let X be a locally Noetherian algebraic spaceover S. The category of coherent OX-modules is abelian. More precisely, the kerneland cokernel of a map of coherent OX-modules are coherent. Any extension ofcoherent sheaves is coherent.

https://stacks.math.columbia.edu/tag/07UA

https://stacks.math.columbia.edu/tag/07UB

https://stacks.math.columbia.edu/tag/07UC


Proof. Choose a scheme U and a surjective étale morphism f : U → X. Pullbackf∗ is an exact functor as it equals a restriction functor, see Properties of Spaces,Equation (26.1.1). By Lemma 12.2 we can check whether an OX -module F iscoherent by checking whether f∗F is coherent. Hence the lemma follows from thecase of schemes which is Cohomology of Schemes, Lemma 9.2. �

Coherent modules form a Serre subcategory of the category of quasi-coherent OX -modules. This does not hold for modules on a general ringed topos.Lemma 12.4.07UD Let S be a scheme. Let X be a locally Noetherian algebraic spaceover S. Let F be a coherent OX-module. Any quasi-coherent submodule of F iscoherent. Any quasi-coherent quotient module of F is coherent.Proof. Choose a scheme U and a surjective étale morphism f : U → X. Pullbackf∗ is an exact functor as it equals a restriction functor, see Properties of Spaces,Equation (26.1.1). By Lemma 12.2 we can check whether an OX -module G iscoherent by checking whether f∗H is coherent. Hence the lemma follows from thecase of schemes which is Cohomology of Schemes, Lemma 9.3. �

Lemma 12.5.07UE Let S be a scheme. Let X be a locally Noetherian algebraic spaceover S,. Let F , G be coherent OX-modules. The OX-modules F ⊗OX

G andHomOX

(F ,G) are coherent.Proof. Via Lemma 12.2 this follows from the result for schemes, see Cohomologyof Schemes, Lemma 9.4. �

Lemma 12.6.07UF Let S be a scheme. Let X be a locally Noetherian algebraic spaceover S. Let F , G be coherent OX-modules. Let ϕ : G → F be a homomorphism ofOX-modules. Let x be a geometric point of X lying over x ∈ |X|.

(1) If Fx = 0 then there exists an open neighbourhood X ′ ⊂ X of x such thatF|X′ = 0.

(2) If ϕx : Gx → Fx is injective, then there exists an open neighbourhood X ′ ⊂X of x such that ϕ|X′ is injective.

(3) If ϕx : Gx → Fx is surjective, then there exists an open neighbourhoodX ′ ⊂ X of x such that ϕ|X′ is surjective.

(4) If ϕx : Gx → Fx is bijective, then there exists an open neighbourhood X ′ ⊂X of x such that ϕ|X′ is an isomorphism.

Proof. Let ϕ : U → X be an étale morphism where U is a scheme and let u ∈ Ube a point mapping to x. By Properties of Spaces, Lemmas 29.4 and 22.1 as wellas More on Algebra, Lemma 45.1 we see that ϕx is injective, surjective, or bijectiveif and only if ϕu : ϕ∗Fu → ϕ∗Gu has the corresponding property. Thus we canapply the schemes version of this lemma to see that (after possibly shrinking U)the map ϕ∗F → ϕ∗G is injective, surjective, or an isomorphism. Let X ′ ⊂ X be theopen subspace corresponding to |ϕ|(|U |) ⊂ |X|, see Properties of Spaces, Lemma4.8. Since {U → X ′} is a covering for the étale topology, we conclude that ϕ|X′ isinjective, surjective, or an isomorphism as desired. Finally, observe that (1) followsfrom (2) by looking at the map F → 0. �

Lemma 12.7.07UG Let S be a scheme. Let X be a locally Noetherian algebraic spaceover S. Let F be a coherent OX-module. Let i : Z → X be the scheme theo-retic support of F and G the quasi-coherent OZ-module such that i∗G = F , seeMorphisms of Spaces, Definition 15.4. Then G is a coherent OZ-module.

https://stacks.math.columbia.edu/tag/07UD

https://stacks.math.columbia.edu/tag/07UE

https://stacks.math.columbia.edu/tag/07UF

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Proof. The statement of the lemma makes sense as a coherent module is in par-ticular of finite type. Moreover, as Z → X is a closed immersion it is locally offinite type and hence Z is locally Noetherian, see Morphisms of Spaces, Lemmas23.7 and 23.5. Finally, as G is of finite type it is a coherent OZ-module by Lemma12.2 �

Lemma 12.8.08AM Let S be a scheme. Let i : Z → X be a closed immersion of locallyNoetherian algebraic spaces over S. Let I ⊂ OX be the quasi-coherent sheaf ofideals cutting out Z. The functor i∗ induces an equivalence between the category ofcoherent OX-modules annihilated by I and the category of coherent OZ-modules.

Proof. The functor is fully faithful by Morphisms of Spaces, Lemma 14.1. Let Fbe a coherent OX -module annihilated by I. By Morphisms of Spaces, Lemma 14.1we can write F = i∗G for some quasi-coherent sheaf G on Z. To check that G iscoherent we can work étale locally (Lemma 12.2). Choosing an étale covering bya scheme we conclude that G is coherent by the case of schemes (Cohomology ofSchemes, Lemma 9.8). Hence the functor is fully faithful and the proof is done. �

Lemma 12.9.07UH Let S be a scheme. Let f : X → Y be a finite morphism ofalgebraic spaces over S with Y locally Noetherian. Let F be a coherent OX-module.Assume f is finite and Y locally Noetherian. Then Rpf∗F = 0 for p > 0 and f∗Fis coherent.

Proof. Choose a scheme V and a surjective étale morphism V → Y . Then V ×YX → V is a finite morphism of locally Noetherian schemes. By (3.0.1) we reduceto the case of schemes which is Cohomology of Schemes, Lemma 9.9. �

13. Coherent sheaves on Noetherian spaces

07UI In this section we mention some properties of coherent sheaves on Noetherian al-gebraic spaces.

Lemma 13.1.07UJ Let S be a scheme. Let X be a Noetherian algebraic space over S.Let F be a coherent OX-module. The ascending chain condition holds for quasi-coherent submodules of F . In other words, given any sequence

F1 ⊂ F2 ⊂ . . . ⊂ F

of quasi-coherent submodules, then Fn = Fn+1 = . . . for some n ≥ 0.

Proof. Choose an affine scheme U and a surjective étale morphism U → X (seeProperties of Spaces, Lemma 6.3). Then U is a Noetherian scheme (by Morphismsof Spaces, Lemma 23.5). If Fn|U = Fn+1|U = . . . then Fn = Fn+1 = . . .. Hencethe result follows from the case of schemes, see Cohomology of Schemes, Lemma10.1. �

Lemma 13.2.07UK Let S be a scheme. Let X be a Noetherian algebraic space over S.Let F be a coherent sheaf on X. Let I ⊂ OX be a quasi-coherent sheaf of idealscorresponding to a closed subspace Z ⊂ X. Then there is some n ≥ 0 such thatInF = 0 if and only if Supp(F) ⊂ Z (set theoretically).

Proof. Choose an affine scheme U and a surjective étale morphism U → X (seeProperties of Spaces, Lemma 6.3). Then U is a Noetherian scheme (by Morphismsof Spaces, Lemma 23.5). Note that InF|U = 0 if and only if InF = 0 and similarly

https://stacks.math.columbia.edu/tag/08AM

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for the condition on the support. Hence the result follows from the case of schemes,see Cohomology of Schemes, Lemma 10.2. �

Lemma 13.3 (Artin-Rees).07UL Let S be a scheme. Let X be a Noetherian algebraicspace over S. Let F be a coherent sheaf on X. Let G ⊂ F be a quasi-coherentsubsheaf. Let I ⊂ OX be a quasi-coherent sheaf of ideals. Then there exists a c ≥ 0such that for all n ≥ c we have

In−c(IcF ∩ G) = InF ∩ G

Proof. Choose an affine scheme U and a surjective étale morphism U → X (seeProperties of Spaces, Lemma 6.3). Then U is a Noetherian scheme (by Morphismsof Spaces, Lemma 23.5). The equality of the lemma holds if and only if it holds afterrestricting to U . Hence the result follows from the case of schemes, see Cohomologyof Schemes, Lemma 10.3. �

Lemma 13.4.07UM Let S be a scheme. Let X be a Noetherian algebraic space overS. Let F be a quasi-coherent OX-module. Let G be a coherent OX-module. LetI ⊂ OX be a quasi-coherent sheaf of ideals. Denote Z ⊂ X the corresponding closedsubspace and set U = X \ Z. There is a canonical isomorphism

colimn HomOX(InG,F) −→ HomOU

(G|U ,F|U ).In particular we have an isomorphism

colimn HomOX(In,F) −→ Γ(U,F).

Proof. Let W be an affine scheme and let W → X be a surjective étale morphism(see Properties of Spaces, Lemma 6.3). Set R = W ×X W . Then W and R areNoetherian schemes, see Morphisms of Spaces, Lemma 23.5. Hence the result holdfor the restrictions of F , G, and I, U , Z to W and R by Cohomology of Schemes,Lemma 10.5. It follows formally that the result holds over X. �

14. Devissage of coherent sheaves

07UN This section is the analogue of Cohomology of Schemes, Section 12.

Lemma 14.1.07UP Let S be a scheme. Let X be a Noetherian algebraic space over S.Let F be a coherent sheaf on X. Suppose that Supp(F) = Z ∪Z ′ with Z, Z ′ closed.Then there exists a short exact sequence of coherent sheaves

0→ G′ → F → G → 0with Supp(G′) ⊂ Z ′ and Supp(G) ⊂ Z.

Proof. Let I ⊂ OX be the sheaf of ideals defining the reduced induced closedsubspace structure on Z, see Properties of Spaces, Lemma 12.3. Consider thesubsheaves G′n = InF and the quotients Gn = F/InF . For each n we have a shortexact sequence

0→ G′n → F → Gn → 0For every geometric point x of Z ′ \Z we have Ix = OX,x and hence Gn,x = 0. Thuswe see that Supp(Gn) ⊂ Z. Note that X \Z ′ is a Noetherian algebraic space. Henceby Lemma 13.2 there exists an n such that G′n|X\Z′ = InF|X\Z′ = 0. For such ann we see that Supp(G′n) ⊂ Z ′. Thus setting G′ = G′n and G = Gn works. �

In the following we will freely use the scheme theoretic support of finite type mod-ules as defined in Morphisms of Spaces, Definition 15.4.

https://stacks.math.columbia.edu/tag/07UL

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Lemma 14.2.07UQ Let S be a scheme. Let X be a Noetherian algebraic space over S.Let F be a coherent sheaf on X. Assume that the scheme theoretic support of F isa reduced Z ⊂ X with |Z| irreducible. Then there exist an integer r > 0, a nonzerosheaf of ideals I ⊂ OZ , and an injective map of coherent sheaves

i∗(I⊕r

)→ F

whose cokernel is supported on a proper closed subspace of Z.

Proof. By assumption there exists a coherent OZ-module G with support Z andF ∼= i∗G, see Lemma 12.7. Hence it suffices to prove the lemma for the case Z = Xand i = id.

By Properties of Spaces, Proposition 13.3 there exists a dense open subspace U ⊂ Xwhich is a scheme. Note that U is a Noetherian integral scheme. After shrinking Uwe may assume that F|U ∼= O⊕rU (for example by Cohomology of Schemes, Lemma12.2 or by a direct algebra argument). Let I ⊂ OX be a quasi-coherent sheafof ideals whose associated closed subspace is the complement of U in X (see forexample Properties of Spaces, Section 12). By Lemma 13.4 there exists an n ≥ 0and a morphism In(O⊕rX ) → F which recovers our isomorphism over U . SinceIn(O⊕rX ) = (In)⊕r we get a map as in the lemma. It is injective: namely, if σ isa nonzero section of I⊕r over a scheme W étale over X, then because X henceW is reduced the support of σ contains a nonempty open of W . But the kernel of(In)⊕r → F is zero over a dense open, hence σ cannot be a section of the kernel. �

Lemma 14.3.07UR Let S be a scheme. Let X be a Noetherian algebraic space over S.Let F be a coherent sheaf on X. There exists a filtration

0 = F0 ⊂ F1 ⊂ . . . ⊂ Fm = F

by coherent subsheaves such that for each j = 1, . . . ,m there exists a reduced closedsubspace Zj ⊂ X with |Zj | irreducible and a sheaf of ideals Ij ⊂ OZj

such that

Fj/Fj−1 ∼= (Zj → X)∗Ij

Proof. Consider the collection

T ={T ⊂ |X| closed such that there exists a coherent sheaf F

with Supp(F) = T for which the lemma is wrong

}We are trying to show that T is empty. If not, then because |X| is Noetherian(Properties of Spaces, Lemma 24.2) we can choose a minimal element T ∈ T . Thismeans that there exists a coherent sheaf F on X whose support is T and for whichthe lemma does not hold. Clearly T 6= ∅ since the only sheaf whose support isempty is the zero sheaf for which the lemma does hold (with m = 0).

If T is not irreducible, then we can write T = Z1∪Z2 with Z1, Z2 closed and strictlysmaller than T . Then we can apply Lemma 14.1 to get a short exact sequence ofcoherent sheaves

0→ G1 → F → G2 → 0with Supp(Gi) ⊂ Zi. By minimality of T each of Gi has a filtration as in thestatement of the lemma. By considering the induced filtration on F we arrive at acontradiction. Hence we conclude that T is irreducible.

https://stacks.math.columbia.edu/tag/07UQ

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Suppose T is irreducible. Let J be the sheaf of ideals defining the reduced inducedclosed subspace structure on T , see Properties of Spaces, Lemma 12.3. By Lemma13.2 we see there exists an n ≥ 0 such that J nF = 0. Hence we obtain a filtration

0 = InF ⊂ In−1F ⊂ . . . ⊂ IF ⊂ F

each of whose successive subquotients is annihilated by J . Hence if each of thesesubquotients has a filtration as in the statement of the lemma then also F does. Inother words we may assume that J does annihilate F .Assume T is irreducible and JF = 0 where J is as above. Then the schemetheoretic support of F is T , see Morphisms of Spaces, Lemma 14.1. Hence we canapply Lemma 14.2. This gives a short exact sequence

0→ i∗(I⊕r)→ F → Q→ 0where the support of Q is a proper closed subset of T . Hence we see that Q hasa filtration of the desired type by minimality of T . But then clearly F does too,which is our final contradiction. �

Lemma 14.4.07US Let S be a scheme. Let X be a Noetherian algebraic space over S.Let P be a property of coherent sheaves on X. Assume

(1) For any short exact sequence of coherent sheaves0→ F1 → F → F2 → 0

if Fi, i = 1, 2 have property P then so does F .(2) For every reduced closed subspace Z ⊂ X with |Z| irreducible and every

quasi-coherent sheaf of ideals I ⊂ OZ we have P for i∗I.Then property P holds for every coherent sheaf on X.

Proof. First note that if F is a coherent sheaf with a filtration0 = F0 ⊂ F1 ⊂ . . . ⊂ Fm = F

by coherent subsheaves such that each of Fi/Fi−1 has property P, then so does F .This follows from the property (1) for P. On the other hand, by Lemma 14.3 we canfilter any F with successive subquotients as in (2). Hence the lemma follows. �

Here is a more useful variant of the lemma above.

Lemma 14.5.07UT Let S be a scheme. Let X be a Noetherian algebraic space over S.Let P be a property of coherent sheaves on X. Assume

(1) For any short exact sequence of coherent sheaves0→ F1 → F → F2 → 0

if Fi, i = 1, 2 have property P then so does F .(2) If P holds for F⊕r for some r ≥ 1, then it holds for F .(3) For every reduced closed subspace i : Z → X with |Z| irreducible there exists

a coherent sheaf G on Z such that(a) Supp(G) = Z,(b) for every nonzero quasi-coherent sheaf of ideals I ⊂ OZ there exists a

quasi-coherent subsheaf G′ ⊂ IG such that Supp(G/G′) is proper closedin |Z| and such that P holds for i∗G′.

Then property P holds for every coherent sheaf on X.

https://stacks.math.columbia.edu/tag/07US

https://stacks.math.columbia.edu/tag/07UT


Proof. Consider the collection

T ={T ⊂ |X| nonempty closed such that there exists a coherent sheaf

F with Supp(F) = T for which the lemma is wrong

}We are trying to show that T is empty. If not, then because |X| is Noetherian(Properties of Spaces, Lemma 24.2) we can choose a minimal element T ∈ T . Thismeans that there exists a coherent sheaf F on X whose support is T and for whichthe lemma does not hold.

If T is not irreducible, then we can write T = Z1∪Z2 with Z1, Z2 closed and strictlysmaller than T . Then we can apply Lemma 14.1 to get a short exact sequence ofcoherent sheaves

0→ G1 → F → G2 → 0with Supp(Gi) ⊂ Zi. By minimality of T each of Gi has P. Hence F has propertyP by (1), a contradiction.

Suppose T is irreducible. Let J be the sheaf of ideals defining the reduced inducedclosed subspace structure on T , see Properties of Spaces, Lemma 12.3. By Lemma13.2 we see there exists an n ≥ 0 such that J nF = 0. Hence we obtain a filtration

0 = J nF ⊂ J n−1F ⊂ . . . ⊂ JF ⊂ F

each of whose successive subquotients is annihilated by J . Hence if each of thesesubquotients has a filtration as in the statement of the lemma then also F does by(1). In other words we may assume that J does annihilate F .

Assume T is irreducible and JF = 0 where J is as above. Denote i : Z → X theclosed subspace corresponding to J . Then F = i∗H for some coherent OZ-moduleH, see Morphisms of Spaces, Lemma 14.1 and Lemma 12.7. Let G be the coherentsheaf on Z satisfying (3)(a) and (3)(b). We apply Lemma 14.2 to get injective maps

I⊕r11 → H and I⊕r22 → G

where the support of the cokernels are proper closed in Z. Hence we find annonempty open V ⊂ Z such that

H⊕r2V∼= G⊕r1V

Let I ⊂ OZ be a quasi-coherent ideal sheaf cutting out Z \ V we obtain (Lemma13.4) a map

InG⊕r1 −→ H⊕r2

which is an isomorphism over V . The kernel is supported on Z\V hence annihilatedby some power of I, see Lemma 13.2. Thus after increasing n we may assume thedisplayed map is injective, see Lemma 13.3. Applying (3)(b) we find G′ ⊂ InG suchthat

(i∗G′)⊕r1 −→ i∗H⊕r2 = F⊕r2

is injective with cokernel supported in a proper closed subset of Z and such thatproperty P holds for i∗G′. By (1) property P holds for (i∗G′)⊕r1 . By (1) andminimality of T = |Z| property P holds for F⊕r2 . And finally by (2) property Pholds for F which is the desired contradiction. �

Lemma 14.6.08AN Let S be a scheme. Let X be a Noetherian algebraic space over S.Let P be a property of coherent sheaves on X. Assume

https://stacks.math.columbia.edu/tag/08AN


(1) For any short exact sequence of coherent sheaves on X if two out of threehave property P so does the third.

(2) If P holds for F⊕r for some r ≥ 1, then it holds for F .(3) For every reduced closed subspace i : Z → X with |Z| irreducible there exists

a coherent sheaf G on X whose scheme theoretic support is Z such that Pholds for G.

Then property P holds for every coherent sheaf on X.

Proof. We will show that conditions (1) and (2) of Lemma 14.4 hold. This is clearfor condition (1). To show that (2) holds, let

T ={i : Z → X reduced closed subspace with |Z| irreducible suchthat i∗I does not have P for some quasi-coherent I ⊂ OZ

}If T is nonempty, then since X is Noetherian, we can find an i : Z → X which isminimal in T . We will show that this leads to a contradiction.Let G be the sheaf whose scheme theoretic support is Z whose existence is assumedin assumption (3). Let ϕ : i∗I⊕r → G be as in Lemma 14.2. Let

0 = F0 ⊂ F1 ⊂ . . . ⊂ Fm = Coker(ϕ)be a filtration as in Lemma 14.3. By minimality of Z and assumption (1) we seethat Coker(ϕ) has property P. As ϕ is injective we conclude using assumption (1)once more that i∗I⊕r has property P. Using assumption (2) we conclude that i∗Ihas property P.Finally, if J ⊂ OZ is a second quasi-coherent sheaf of ideals, set K = I ∩ J andconsider the short exact sequences

0→ K → I → I/K → 0 and 0→ K → J → J /K → 0Arguing as above, using the minimality of Z, we see that i∗I/K and i∗J /K satisfyP. Hence by assumption (1) we conclude that i∗K and then i∗J satisfy P. In otherwords, Z is not an element of T which is the desired contradiction. �

15. Limits of coherent modules

07UU A colimit of coherent modules (on a locally Noetherian algebraic space) is typicallynot coherent. But it is quasi-coherent as any colimit of quasi-coherent moduleson an algebraic space is quasi-coherent, see Properties of Spaces, Lemma 29.7.Conversely, if the algebraic space is Noetherian, then every quasi-coherent moduleis a filtered colimit of coherent modules.

Lemma 15.1.07UV Let S be a scheme. Let X be a Noetherian algebraic space over S.Every quasi-coherent OX-module is the filtered colimit of its coherent submodules.

Proof. Let F be a quasi-coherent OX -module. If G,H ⊂ F are coherent OX -submodules then the image of G ⊕ H → F is another coherent OX -submodulewhich contains both of them (see Lemmas 12.3 and 12.4). In this way we see thatthe system is directed. Hence it now suffices to show that F can be written asa filtered colimit of coherent modules, as then we can take the images of thesemodules in F to conclude there are enough of them.Let U be an affine scheme and U → X a surjective étale morphism. Set R =U ×X U so that X = U/R as usual. By Properties of Spaces, Proposition 32.1we see that QCoh(OX) = QCoh(U,R, s, t, c). Hence we reduce to showing the

https://stacks.math.columbia.edu/tag/07UV


corresponding thing for QCoh(U,R, s, t, c). Thus the result follows from the moregeneral Groupoids, Lemma 15.4. �

Lemma 15.2.07UW Let S be a scheme. Let f : X → Y be an affine morphism ofalgebraic spaces over S with Y Noetherian. Then every quasi-coherent OX-moduleis a filtered colimit of finitely presented OX-modules.

Proof. Let F be a quasi-coherent OX -module. Write f∗F = colimHi with Hi acoherent OY -module, see Lemma 15.1. By Lemma 12.2 the modules Hi are OY -modules of finite presentation. Hence f∗Hi is an OX -module of finite presentation,see Properties of Spaces, Section 30. We claim the map

colim f∗Hi = f∗f∗F → F

is surjective as f is assumed affine, Namely, choose a scheme V and a surjectiveétale morphism V → Y . Set U = X ×Y V . Then U is a scheme, f ′ : U → Vis affine, and U → X is surjective étale. By Properties of Spaces, Lemma 26.2we see that f ′∗(F|U ) = f∗F|V and similarly for pullbacks. Thus the restriction off∗f∗F → F to U is the map

f∗f∗F|U = (f ′)∗(f∗F)|V ) = (f ′)∗f ′∗(F|U )→ F|U

which is surjective as f ′ is an affine morphism of schemes. Hence the claim holds.

We conclude that every quasi-coherent module on X is a quotient of a filteredcolimit of finitely presented modules. In particular, we see that F is a cokernel ofa map

colimj∈J Gj −→ colimi∈I Hiwith Gj and Hi finitely presented. Note that for every j ∈ I there exist i ∈ I anda morphism α : Gj → Hi such that

Gj α//

��

Hi

��colimj∈J Gj // colimi∈I Hi

commutes, see Lemma 5.3. In this situation Coker(α) is a finitely presented OX -module which comes endowed with a map Coker(α) → F . Consider the set K oftriples (i, j, α) as above. We say that (i, j, α) ≤ (i′, j′, α′) if and only if i ≤ i′,j ≤ j′, and the diagram

Gj α//

��

Hi

��Gj′

α′ // Hi′

commutes. It follows from the above that K is a directed partially ordered set,

F = colim(i,j,α)∈K Coker(α),

and we win. �

https://stacks.math.columbia.edu/tag/07UW


16. Vanishing of cohomology

07UX In this section we show that a quasi-compact and quasi-separated algebraic spaceis affine if it has vanishing higher cohomology for all quasi-coherent sheaves. Wedo this in a sequence of lemmas all of which will become obsolete once we proveProposition 16.7.

Situation 16.1.07UY Here S is a scheme and X is a quasi-compact and quasi-separatedalgebraic space over S with the following property: For every quasi-coherent OX -module F we have H1(X,F) = 0. We set A = Γ(X,OX).

We would like to show that the canonical morphism

p : X −→ Spec(A)

(see Properties of Spaces, Lemma 33.1) is an isomorphism. If M is an A-modulewe denote M ⊗A OX the quasi-coherent module p∗M .

Lemma 16.2.07UZ In Situation 16.1 for an A-module M we have p∗(M ⊗AOX) = Mand Γ(X,M ⊗A OX) = M .

Proof. The equality p∗(M ⊗A OX) = M follows from the equality Γ(X,M ⊗AOX) = M as p∗(M ⊗A OX) is a quasi-coherent module on Spec(A) by Morphismsof Spaces, Lemma 11.2. Observe that Γ(X,

⊕i∈I OX) =

⊕i∈I A by Lemma 5.1.

Hence the lemma holds for free modules. Choose a short exact sequence F1 →F0 → M where F0, F1 are free A-modules. Since H1(X,−) is zero the globalsections functor is right exact. Moreover the pullback p∗ is right exact as well.Hence we see that

Γ(X,F1 ⊗A OX)→ Γ(X,F0 ⊗A OX)→ Γ(X,M ⊗A OX)→ 0

is exact. The result follows. �

The following lemma shows that Situation 16.1 is preserved by base change ofX → Spec(A) by Spec(A′)→ Spec(A).

Lemma 16.3.07V0 In Situation 16.1.(1) Given an affine morphism X ′ → X of algebraic spaces, we have H1(X ′,F ′) =

0 for every quasi-coherent OX′-module F ′.(2) Given an A-algebra A′ setting X ′ = X ×Spec(A) Spec(A′) the morphism

X ′ → X is affine and Γ(X ′,OX′) = A′.

Proof. Part (1) follows from Lemma 8.2 and the Leray spectral sequence (Coho-mology on Sites, Lemma 14.5). Let A → A′ be as in (2). Then X ′ → X is affinebecause affine morphisms are preserved under base change (Morphisms of Spaces,Lemma 20.5) and the fact that a morphism of affine schemes is affine. The equalityΓ(X ′,OX′) = A′ follows as (X ′ → X)∗OX′ = A′ ⊗A OX by Lemma 11.1 and thus

Γ(X ′,OX′) = Γ(X, (X ′ → X)∗OX′) = Γ(X,A′ ⊗A OX) = A′

by Lemma 16.2. �

Lemma 16.4.07V1 In Situation 16.1. Let Z0, Z1 ⊂ |X| be disjoint closed subsets.Then there exists an a ∈ A such that Z0 ⊂ V (a) and Z1 ⊂ V (a− 1).

https://stacks.math.columbia.edu/tag/07UY

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Proof. We may and do endow Z0, Z1 with the reduced induced subspace structure(Properties of Spaces, Definition 12.5) and we denote i0 : Z0 → X and i1 : Z1 → Xthe corresponding closed immersions. Since Z0 ∩ Z1 = ∅ we see that the canonicalmap of quasi-coherent OX -modules

OX −→ i0,∗OZ0 ⊕ i1,∗OZ1

is surjective (look at stalks at geometric points). Since H1(X,−) is zero on thekernel of this map the induced map of global sections is surjective. Thus we canfind a ∈ A which maps to the global section (0, 1) of the right hand side. �

Lemma 16.5.07V4 In Situation 16.1 the morphism p : X → Spec(A) is universallyinjective.

Proof. Let A→ k be a ring homomorphism where k is a field. It suffices to showthat Spec(k)×Spec(A) X has at most one point (see Morphisms of Spaces, Lemma19.6). Using Lemma 16.3 we may assume that A is a field and we have to showthat |X| has at most one point.

Let’s think of X as an algebraic space over Spec(k) and let’s use the notation X(K)to denote K-valued points of X for any extension K/k, see Morphisms of Spaces,Section 24. If K/k is an algebraically closed field extension of large transcendencedegree, then we see that X(K) → |X| is surjective, see Morphisms of Spaces,Lemma 24.2. Hence, after replacing k by K, we see that it suffices to prove thatX(k) is a singleton (in the case A = k).

Let x, x′ ∈ X(k). By Decent Spaces, Lemma 14.4 we see that x and x′ are closedpoints of |X|. Hence x and x′ map to distinct points of Spec(k) if x 6= x′ by Lemma16.4. We conclude that x = x′ as desired. �

Lemma 16.6.07V5 In Situation 16.1 the morphism p : X → Spec(A) is separated.

Proof. By Decent Spaces, Lemma 9.2 we can find a scheme Y and a surjectiveintegral morphism Y → X. Since an integral morphism is affine, we can applyLemma 16.3 to see that H1(Y,G) = 0 for every quasi-coherent OY -module G. SinceY → X is quasi-compact and X is quasi-compact, we see that Y is quasi-compact.Since Y is a scheme, we may apply Cohomology of Schemes, Lemma 3.1 to seethat Y is affine. Hence Y is separated. Note that an integral morphism is affineand universally closed, see Morphisms of Spaces, Lemma 45.7. By Morphisms ofSpaces, Lemma 9.8 we see that X is a separated algebraic space. �

Proposition 16.7.07V6 A quasi-compact and quasi-separated algebraic space is affineif and only if all higher cohomology groups of quasi-coherent sheaves vanish. Moreprecisely, any algebraic space as in Situation 16.1 is an affine scheme.

Proof. Choose an affine scheme U = Spec(B) and a surjective étale morphismϕ : U → X. Set R = U ×X U . As p is separated (Lemma 16.6) we see that R is aclosed subscheme of U ×Spec(A) U = Spec(B ⊗A B). Hence R = Spec(C) is affinetoo and the ring map

B ⊗A B −→ C

is surjective. Let us denote the two maps s, t : B → C as usual. Pick g1, . . . , gm ∈ Bsuch that s(g1), . . . , s(gm) generate C over t : B → C (which is possible as t : B → C





is of finite presentation and the displayed map is surjective). Then g1, . . . , gm giveglobal sections of ϕ∗OU and the map

OX [z1, . . . , zn] −→ ϕ∗OU , zj 7−→ gj

is surjective: you can check this by restricting to U . Namely, ϕ∗ϕ∗OU = t∗OR(by Lemma 11.2) hence you get exactly the condition that s(gi) generate C overt : B → C. By the vanishing of H1 of the kernel we see that

Γ(X,OX [x1, . . . , xn]) = A[x1, . . . , xn] −→ Γ(X,ϕ∗OU ) = Γ(U,OU ) = B

is surjective. Thus we conclude that B is a finite type A-algebra. Hence X →Spec(A) is of finite type and separated. By Lemma 16.5 and Morphisms of Spaces,Lemma 27.5 it is also locally quasi-finite. Hence X → Spec(A) is representable byMorphisms of Spaces, Lemma 51.1 and X is a scheme. Finally X is affine, henceequal to Spec(A), by an application of Cohomology of Schemes, Lemma 3.1. �

Lemma 16.8.0D2V Let S be a scheme. Let X be a Noetherian algebraic space over S.Assume that for every coherent OX-module F we have H1(X,F) = 0. Then X isan affine scheme.

Proof. The assumption implies that H1(X,F) = 0 for every quasi-coherent OX -module F by Lemmas 15.1 and 5.1. Then X is affine by Proposition 16.7. �

Lemma 16.9.0D2W Let S be a scheme. Let X be a Noetherian algebraic space over S.Let L be an invertible OX-module. Assume that for every coherent OX-module Fthere exists an n ≥ 1 such that H1(X,F ⊗OX

L⊗n) = 0. Then X is a scheme andL is ample on X.

Proof. Let s ∈ H0(X,L⊗d) be a global section. Let U ⊂ X be the open subspaceover which s is a generator of L⊗d. In particular we have L⊗d|U ∼= OU . We claimthat U is affine.

Proof of the claim. We will show that H1(U,F) = 0 for every quasi-coherent OU -module F . This will prove the claim by Proposition 16.7. Denote j : U → X theinclusion morphism. Since étale locally the morphism j is affine (by Morphisms,Lemma 11.10) we see that j is affine (Morphisms of Spaces, Lemma 20.3). Hencewe have

H1(U,F) = H1(X, j∗F)

by Lemma 8.2 (and Cohomology on Sites, Lemma 14.6). Write j∗F = colimFi asa filtered colimit of coherent OX -modules, see Lemma 15.1. Then

H1(X, j∗F) = colimH1(X,Fi)

by Lemma 5.1. Thus it suffices to show thatH1(X,Fi) maps to zero inH1(U, j∗Fi).By assumption there exists an n ≥ 1 such that

H1(X,Fi ⊗OX(OX ⊕ L⊕ . . .⊕ L⊗d−1)⊗OX

L⊗n) = 0

Hence there exists an a ≥ 0 such that H1(X,Fi ⊗OXL⊗ad) = 0. On the other

hand, the mapsa : Fi −→ Fi ⊗OX

L⊗ad

https://stacks.math.columbia.edu/tag/0D2V

https://stacks.math.columbia.edu/tag/0D2W


is an isomorphism after restriction to U . Contemplating the commutative diagram

H1(X,Fi) //

sa

��

H1(U, j∗Fi)

∼=��

H1(X,Fi ⊗OXL⊗ad) // H1(U, j∗(Fi ⊗OX

L⊗ad))

we conclude that the map H1(X,Fi)→ H1(U, j∗Fi) is zero and the claim holds.

Let x ∈ |X| be a closed point. By Decent Spaces, Lemma 14.6 we can represent x bya closed immersion i : Spec(k)→ X (this also uses that a quasi-separated algebraicspace is decent, see Decent Spaces, Section 6). Thus OX → i∗OSpec(k) is surjective.Let I ⊂ OX be the kernel and choose d ≥ 1 such that H1(X, I ⊗OX

L⊗d) = 0.Then

H0(X,L⊗d)→ H0(X, i∗OSpec(k) ⊗OXL⊗d) = H0(Spec(k), i∗L⊗d) ∼= k

is surjective by the long exact cohomology sequence. Hence there exists an s ∈H0(X,L⊗d) such that x ∈ U where U is the open subspace corresponding to s asabove. Thus x is in the schematic locus (see Properties of Spaces, Lemma 13.1) ofX by our claim.

To conclude that X is a scheme, it suffices to show that any open subset of |X|which contains all the closed points is equal to |X|. This follows from the factthat |X| is a Noetherian topological space, see Properties of Spaces, Lemma 24.3.Finally, if X is a scheme, then we can apply Cohomology of Schemes, Lemma 3.3to conclude that L is ample. �

17. Finite morphisms and affines

07VN This section is the analogue of Cohomology of Schemes, Section 13.

Lemma 17.1.0GF7 Let S be a scheme. Let f : Y → X be a morphism of algebraicspaces over S. Assume f is finite, surjective and X locally Noetherian. Let i : Z →X be a closed immersion. Denote i′ : Z ′ → Y the inverse image of Z (Morphismsof Spaces, Section 13) and f ′ : Z ′ → Z the induced morphism. Then G = f ′∗OZ′ isa coherent OZ-module whose support is Z.

Proof. Observe that f ′ is the base change of f and hence is finite and surjectiveby Morphisms of Spaces, Lemmas 5.5 and 45.5. Note that Y , Z, and Z ′ arelocally Noetherian by Morphisms of Spaces, Lemma 23.5 (and the fact that closedimmersions and finite morphisms are of finite type). By Lemma 12.9 we see thatG is a coherent OZ-module. The support of G is closed in |Z|, see Morphisms ofSpaces, Lemma 15.2. Hence if the support of G is not equal to |Z|, then afterreplacing X by an open subspace we may assume G = 0 but Z 6= ∅. This wouldmean that f ′∗OZ′ = 0. In particular the section 1 ∈ Γ(Z ′,OZ′) = Γ(Z, f ′∗OZ′)would be zero which would imply Z ′ = ∅ is the empty algebraic space. This isimpossible as Z ′ → Z is surjective. �

Lemma 17.2.0GF8 Let S be a scheme. Let f : Y → X be a morphism of algebraicspaces over S. Let F be a quasi-coherent sheaf on Y . Let I be a quasi-coherentsheaf of ideals on X. If f is affine then If∗F = f∗(f−1IF) (with notation asexplained in the proof).

https://stacks.math.columbia.edu/tag/0GF7

https://stacks.math.columbia.edu/tag/0GF8


Proof. The notation means the following. Since f−1 is an exact functor we seethat f−1I is a sheaf of ideals of f−1OX . Via the map f ] : f−1OX → OY on Yetalethis acts on F . Then f−1IF is the subsheaf generated by sums of local sections ofthe form as where a is a local section of f−1I and s is a local section of F . It isa quasi-coherent OY -submodule of F because it is also the image of a natural mapf∗I ⊗OY

F → F .

Having said this the proof is straightforward. Namely, the question is étale localon X and hence we may assume X is an affine scheme. In this case the result is aconsequence of the corresponding result for schemes, see Cohomology of Schemes,Lemma 13.2. �

Lemma 17.3.07VP Let S be a scheme. Let f : Y → X be a morphism of algebraicspaces over S. Assume

(1) f finite,(2) f surjective,(3) Y affine, and(4) X Noetherian.

Then X is affine.

Proof. We will prove that under the assumptions of the lemma for any coherentOX -module F we have H1(X,F) = 0. This implies that H1(X,F) = 0 for everyquasi-coherent OX -module F by Lemmas 15.1 and 5.1. Then it follows that X isaffine from Proposition 16.7.

Let P be the property of coherent sheaves F on X defined by the rule

P(F)⇔ H1(X,F) = 0.

We are going to apply Lemma 14.5. Thus we have to verify (1), (2) and (3) ofthat lemma for P. Property (1) follows from the long exact cohomology sequenceassociated to a short exact sequence of sheaves. Property (2) follows sinceH1(X,−)is an additive functor. To see (3) let i : Z → X be a reduced closed subspace with|Z| irreducible. Let i′ : Z ′ → Y and f ′ : Z ′ → Z be as in Lemma 17.1 and setG = f ′∗OZ′ . We claim that G satisfies properties (3)(a) and (3)(b) of Lemma 14.5which will finish the proof. Property (3)(a) we have seen in Lemma 17.1. To see(3)(b) let I be a nonzero quasi-coherent sheaf of ideals on Z. Denote I ′ ⊂ OZ′the quasi-coherent ideal (f ′)−1IOZ′ , i.e., the image of (f ′)∗I → OZ′ . By Lemma17.2 we have f∗I ′ = IG. We claim the common value G′ = IG = f ′∗I ′ satisfies thecondition expressed in (3)(b). First, it is clear that the support of G/G′ is containedin the support of OZ/I which is a proper subspace of |Z| as I is a nonzero idealsheaf on the reduced and irreducible algebraic space Z. The morphism f ′ is affine,hence R1f ′∗I ′ = 0 by Lemma 8.2. As Z ′ is affine (as a closed subscheme of an affinescheme) we have H1(Z ′, I ′) = 0. Hence the Leray spectral sequence (in the formCohomology on Sites, Lemma 14.6) implies that H1(Z, f ′∗I ′) = 0. Since i : Z → Xis affine we conclude that R1i∗f

′∗I ′ = 0 hence H1(X, i∗f ′∗I ′) = 0 by Leray again.

In other words, we have H1(X, i∗G′) = 0 as desired. �

18. A weak version of Chow’s lemma

089I In this section we quickly prove the following lemma in order to help us prove thebasic results on cohomology of coherent modules on proper algebraic spaces.

https://stacks.math.columbia.edu/tag/07VP


Lemma 18.1.089J Let A be a ring. Let X be an algebraic space over Spec(A) whosestructure morphism X → Spec(A) is separated of finite type. Then there existsa proper surjective morphism X ′ → X where X ′ is a scheme which is H-quasi-projective over Spec(A).

Proof. Let W be an affine scheme and let f : W → X be a surjective étale mor-phism. There exists an integer d such that all geometric fibres of f have ≤ d points(because X is a separated algebraic hence reasonable, see Decent Spaces, Lemma5.1). Picking d minimal we get a nonempty open U ⊂ X such that f−1(U)→ U isfinite étale of degree d, see Decent Spaces, Lemma 8.1. Let

V ⊂W ×X W ×X . . .×X W

(d factors in the fibre product) be the complement of all the diagonals. BecauseW → X is separated the diagonal W → W ×X W is a closed immersion. SinceW → X is étale the diagonal W →W ×XW is an open immersion, see Morphismsof Spaces, Lemmas 39.10 and 38.9. Hence the diagonals are open and closed sub-schemes of the quasi-compact schemeW ×X . . .×XW . In particular we conclude Vis a quasi-compact scheme. Choose an open immersionW ⊂ Y with Y H-projectiveover A (this is possible as W is affine and of finite type over A; for example we canuse Morphisms, Lemmas 39.2 and 43.11). Let

Z ⊂ Y ×A Y ×A . . .×A Y

be the scheme theoretic image of the composition V → W ×X . . . ×X W → Y ×A. . . ×A Y . Observe that this morphism is quasi-compact since V is quasi-compactand Y ×A . . . ×A Y is separated. Note that V → Z is an open immersion asV → Y ×A . . .×A Y is an immersion, see Morphisms, Lemma 7.7. The projectionmorphisms give d morphisms gi : Z → Y . These morphisms gi are projective as Yis projective over A, see material in Morphisms, Section 43. We set

X ′ =⋃g−1i (W ) ⊂ Z

There is a morphism X ′ → X whose restriction to g−1i (W ) is the composition

g−1i (W ) → W → X. Namely, these morphisms agree over V hence agree overg−1i (W ) ∩ g−1

j (W ) by Morphisms of Spaces, Lemma 17.8. Claim: the morphismX ′ → X is proper.

If the claim holds, then the lemma follows by induction on d. Namely, by construc-tion X ′ is H-quasi-projective over Spec(A). The image of X ′ → X contains theopen U as V surjects onto U . Denote T the reduced induced algebraic space struc-ture on X \U . Then T ×X W is a closed subscheme of W , hence affine. Moreover,the morphism T ×X W → T is étale and every geometric fibre has < d points. Byinduction hypothesis there exists a proper surjective morphism T ′ → T where T ′ isa scheme H-quasi-projective over Spec(A). Since T is a closed subspace of X we seethat T ′ → X is a proper morphism. Thus the lemma follows by taking the propersurjective morphism X ′ q T ′ → X.

Proof of the claim. By construction the morphism X ′ → X is separated and offinite type. We will check conditions (1) – (4) of Morphisms of Spaces, Lemma 42.5for the morphisms V → X ′ and X ′ → X. Conditions (1) and (2) we have seenabove. Condition (3) holds as X ′ → X is separated (as a morphism whose source is

https://stacks.math.columbia.edu/tag/089J


a separated algebraic space). Thus it suffices to check liftability to X ′ for diagrams

Spec(K) //

��

V

��Spec(R) // X

where R is a valuation ring with fraction field K. Note that the top horizontal mapis given by d pairwise distinct K-valued points w1, . . . , wd of W . In fact, this is acomplete set of inverse images of the point x ∈ X(K) coming from the diagram.Since W → X is surjective, we can, after possibly replacing R by an extension ofvaluation rings, lift the morphism Spec(R)→ X to a morphism w : Spec(R)→W ,see Morphisms of Spaces, Lemma 42.4. Since w1, . . . , wd is a complete collection ofinverse images of x we see that w|Spec(K) is equal to one of them, say wi. Thus wesee that we get a commutative diagram

Spec(K) //

��

Z

gi

��Spec(R) w // Y

By the valuative criterion of properness for the projective morphism gi we can liftw to z : Spec(R)→ Z, see Morphisms, Lemma 43.5 and Schemes, Proposition 20.6.The image of z is in g−1

i (W ) ⊂ X ′ and the proof is complete. �

19. Noetherian valuative criterion

0ARI We prove a version of the valuative criterion for properness using discrete valuationrings. More precise (and therefore more technical) versions can be found in Limitsof Spaces, Section 21.

Lemma 19.1.0ARJ Let S be a scheme. Let f : X → Y be a morphism of algebraicspaces over S. Assume

(1) Y is locally Noetherian,(2) f is locally of finite type and quasi-separated,(3) for every commutative diagram

Spec(K) //

��

X

��Spec(A) //

;;

Y

where A is a discrete valuation ring and K its fraction field, there is atmost one dotted arrow making the diagram commute.

Then f is separated.

Proof. To prove f is separated, we may work étale locally on Y (Morphisms ofSpaces, Lemma 4.12). Choose an affine scheme U and an étale morphism U →

https://stacks.math.columbia.edu/tag/0ARJ


X ×Y X. Set V = X ×∆,X×Y X U which is quasi-compact because f is quasi-separated. Consider a commutative diagram

Spec(K) //

��

V

��Spec(A) //

;;

U

We can interpret the composition Spec(A)→ U → X×Y X as a pair of morphismsa, b : Spec(A) → X agreeing as morphisms into Y and equal when restricted toSpec(K). Hence our assumption (3) guarantees a = b and we find the dotted arrowin the diagram. By Limits, Lemma 15.3 we conclude that V → U is proper. Inother words, ∆ is proper. Since ∆ is a monomorphism, we find that ∆ is a closedimmersion (Étale Morphisms, Lemma 7.2) as desired. �

Lemma 19.2.0ARK Let S be a scheme. Let f : X → Y be a morphism of algebraicspaces over S. Assume

(1) Y is locally Noetherian,(2) f is of finite type and quasi-separated,(3) for every commutative diagram

Spec(K) //

��

X

��Spec(A) //

;;

Y

where A is a discrete valuation ring and K its fraction field, there is aunique dotted arrow making the diagram commute.

Then f is proper.

Proof. It suffices to prove f is universally closed because f is separated by Lemma19.1. To do this we may work étale locally on Y (Morphisms of Spaces, Lemma9.5). Hence we may assume Y = Spec(A) is a Noetherian affine scheme. ChooseX ′ → X as in the weak form of Chow’s lemma (Lemma 18.1). We claim thatX ′ → Spec(A) is universally closed. The claim implies the lemma by Morphismsof Spaces, Lemma 40.7. To prove this, according to Limits, Lemma 15.4 it sufficesto prove that in every solid commutative diagram

Spec(K) //

��

X ′ // X

��Spec(A) //

a

;;

b

66

Y

whereA is a dvr with fraction fieldK we can find the dotted arrow a. By assumptionwe can find the dotted arrow b. Then the morphism X ′ ×X,b Spec(A) → Spec(A)is a proper morphism of schemes and by the valuative criterion for morphisms ofschemes we can lift b to the desired morphism a. �

Remark 19.3 (Variant for complete discrete valuation rings).0ARL In Lemmas 19.1and 19.2 it suffices to consider complete discrete valuation rings. To be precise in

https://stacks.math.columbia.edu/tag/0ARK

https://stacks.math.columbia.edu/tag/0ARL


Lemma 19.1 we can replace condition (3) by the following condition: Given anycommutative diagram

Spec(K) //

��

X

��Spec(A) //

;;

Y

where A is a complete discrete valuation ring with fraction field K there exists atmost one dotted arrow making the diagram commute. Namely, given any diagramas in Lemma 19.1 (3) the completion A∧ is a discrete valuation ring (More onAlgebra, Lemma 43.5) and the uniqueness of the arrow Spec(A∧) → X impliesthe uniqueness of the arrow Spec(A) → X for example by Properties of Spaces,Proposition 17.1. Similarly in Lemma 19.2 we can replace condition (3) by thefollowing condition: Given any commutative diagram

Spec(K) //

��

X

��Spec(A) // Y

where A is a complete discrete valuation ring with fraction field K there existsan extension A ⊂ A′ of complete discrete valuation rings inducing a fraction fieldextension K ⊂ K ′ such that there exists a unique arrow Spec(A′)→ X making thediagram

Spec(K ′) //

��

Spec(K) // X

��Spec(A′) //

44

Spec(A) // Y

commute. Namely, given any diagram as in Lemma 19.2 part (3) the existence ofany commutative diagram

Spec(L) //

��

Spec(K) // X

��Spec(B) //

44

Spec(A) // Y

for any extension A ⊂ B of discrete valuation rings will imply there exists an arrowSpec(A) → X fitting into the diagram. This was shown in Morphisms of Spaces,Lemma 41.4. In fact, it follows from these considerations that it suffices to lookfor dotted arrows in diagrams for any class of discrete valuation rings such that,given any discrete valuation ring, there is an extension of it that is in the class. Forexample, we could take complete discrete valuation rings with algebraically closedresidue field.

20. Higher direct images of coherent sheaves

08AP In this section we prove the fundamental fact that the higher direct images of acoherent sheaf under a proper morphism are coherent. First we prove a helperlemma.


Lemma 20.1.08AQ Let S be a scheme. Consider a commutative diagram

Xi//

f

PnY

��Y

of algebraic spaces over S. Assume i is a closed immersion and Y Noetherian. SetL = i∗OPn

Y(1). Let F be a coherent module on X. Then there exists an integer d0

such that for all d ≥ d0 we have Rpf∗(F ⊗OXL⊗d) = 0 for all p > 0.

Proof. Checking whether Rpf∗(F⊗L⊗d) is zero can be done étale locally on Y , seeEquation (3.0.1). Hence we may assume Y is the spectrum of a Noetherian ring.In this case X is a scheme and the result follows from Cohomology of Schemes,Lemma 16.2. �

Lemma 20.2.08AR Let S be a scheme. Let f : X → Y be a proper morphism ofalgebraic spaces over S with Y locally Noetherian. Let F be a coherent OX-module.Then Rif∗F is a coherent OY -module for all i ≥ 0.

Proof. We first remark that X is a locally Noetherian algebraic space by Mor-phisms of Spaces, Lemma 23.5. Hence the statement of the lemma makes sense.Moreover, computing Rif∗F commutes with étale localization on Y (Properties ofSpaces, Lemma 26.2) and checking whether Rif∗F coherent can be done étale lo-cally on Y (Lemma 12.2). Hence we may assume that Y = Spec(A) is a Noetherianaffine scheme.Assume Y = Spec(A) is an affine scheme. Note that f is locally of finite presentation(Morphisms of Spaces, Lemma 28.7). Thus it is of finite presentation, hence X isNoetherian (Morphisms of Spaces, Lemma 28.6). Thus Lemma 14.6 applies to thecategory of coherent modules of X. For a coherent sheaf F on X we say P holds ifand only if Rif∗F is a coherent module on Spec(A). We will show that conditions(1), (2), and (3) of Lemma 14.6 hold for this property thereby finishing the proofof the lemma.Verification of condition (1). Let

0→ F1 → F2 → F3 → 0be a short exact sequence of coherent sheaves on X. Consider the long exactsequence of higher direct images

Rp−1f∗F3 → Rpf∗F1 → Rpf∗F2 → Rpf∗F3 → Rp+1f∗F1

Then it is clear that if 2-out-of-3 of the sheaves Fi have property P, then thehigher direct images of the third are sandwiched in this exact complex between twocoherent sheaves. Hence these higher direct images are also coherent by Lemmas12.3 and 12.4. Hence property P holds for the third as well.Verification of condition (2). This follows immediately from the fact that Rif∗(F1⊕F2) = Rif∗F1⊕Rif∗F2 and that a summand of a coherent module is coherent (seelemmas cited above).Verification of condition (3). Let i : Z → X be a closed immersion with Z reducedand |Z| irreducible. Set g = f ◦ i : Z → Spec(A). Let G be a coherent module on Zwhose scheme theoretic support is equal to Z such that Rpg∗G is coherent for all p.

https://stacks.math.columbia.edu/tag/08AQ

https://stacks.math.columbia.edu/tag/08AR


Then F = i∗G is a coherent module on X whose scheme theoretic support is Z suchthat Rpf∗F = Rpg∗G. To see this use the Leray spectral sequence (Cohomology onSites, Lemma 14.7) and the fact that Rqi∗G = 0 for q > 0 by Lemma 8.2 and thefact that a closed immersion is affine. (Morphisms of Spaces, Lemma 20.6). Thuswe reduce to finding a coherent sheaf G on Z with support equal to Z such thatRpg∗G is coherent for all p.

We apply Lemma 18.1 to the morphism Z → Spec(A). Thus we get a diagram

Z

g##

Z ′

g′

��

πoo

i// Pn

A

{{Spec(A)

with π : Z ′ → Z proper surjective and i an immersion. Since Z → Spec(A) isproper we conclude that g′ is proper (Morphisms of Spaces, Lemma 40.4). Hence iis a closed immersion (Morphisms of Spaces, Lemmas 40.6 and 12.3). It follows thatthe morphism i′ = (i, π) : Pn

A ×Spec(A) Z′ = Pn

Z is a closed immersion (Morphismsof Spaces, Lemma 4.6). Set

L = i∗OPnA

(1) = (i′)∗OPnZ

(1)

We may apply Lemma 20.1 to L and π as well as L and g′. Hence for all d� 0 wehave Rpπ∗L⊗d = 0 for all p > 0 and Rp(g′)∗L⊗d = 0 for all p > 0. Set G = π∗L⊗d.By the Leray spectral sequence (Cohomology on Sites, Lemma 14.7) we have

Ep,q2 = Rpg∗Rqπ∗L⊗d ⇒ Rp+q(g′)∗L⊗d

and by choice of d the only nonzero terms in Ep,q2 are those with q = 0 and theonly nonzero terms of Rp+q(g′)∗L⊗d are those with p = q = 0. This implies thatRpg∗G = 0 for p > 0 and that g∗G = (g′)∗L⊗d. Applying Cohomology of Schemes,Lemma 16.3 we see that g∗G = (g′)∗L⊗d is coherent.

We still have to check that the support of G is Z. This follows from the fact that L⊗dhas lots of global sections. We spell it out here. Note that L⊗d is globally generatedfor all d ≥ 0 because the same is true for OPn(d). Pick a point z ∈ Z ′ mapping tothe generic point ξ of Z which we can do as π is surjective. (Observe that Z doesindeed have a generic point as |Z| is irreducible and Z is Noetherian, hence quasi-separated, hence |Z| is a sober topological space by Properties of Spaces, Lemma15.1.) Pick s ∈ Γ(Z ′,L⊗d) which does not vanish at z. Since Γ(Z,G) = Γ(Z ′,L⊗d)we may think of s as a global section of G. Choose a geometric point z of Z ′ lyingover z and denote ξ = g′◦z the corresponding geometric point of Z. The adjunctionmap

(g′)∗G = (g′)∗g′∗L⊗d −→ L⊗d

induces a map of stalks Gξ → Lz, see Properties of Spaces, Lemma 29.5. Moreoverthe adjunction map sends the pullback of s (viewed as a section of G) to s (viewedas a section of L⊗d). Thus the image of s in the vector space which is the sourceof the arrow

Gξ ⊗ κ(ξ) −→ L⊗dz ⊗ κ(z)isn’t zero since by choice of s the image in the target of the arrow is nonzero. Henceξ is in the support of G (Morphisms of Spaces, Lemma 15.2). Since |Z| is irreducible


and Z is reduced we conclude that the scheme theoretic support of G is all of Z asdesired. �

Lemma 20.3.08AS Let A be a Noetherian ring. Let f : X → Spec(A) be a propermorphism of algebraic spaces. Let F be a coherent OX-module. Then Hi(X,F) isfinite A-module for all i ≥ 0.

Proof. This is just the affine case of Lemma 20.2. Namely, by Lemma 3.1 weknow that Rif∗F is a quasi-coherent sheaf. Hence it is the quasi-coherent sheafassociated to the A-module Γ(Spec(A), Rif∗F) = Hi(X,F). The equality holds byCohomology on Sites, Lemma 14.6 and vanishing of higher cohomology groups ofquasi-coherent modules on affine schemes (Cohomology of Schemes, Lemma 2.2).By Lemma 12.2 we see Rif∗F is a coherent sheaf if and only if Hi(X,F) is anA-module of finite type. Hence Lemma 20.2 gives us the conclusion. �

Lemma 20.4.08AT Let A be a Noetherian ring. Let B be a finitely generated gradedA-algebra. Let f : X → Spec(A) be a proper morphism of algebraic spaces. SetB = f∗B. Let F be a quasi-coherent graded B-module of finite type. For everyp ≥ 0 the graded B-module Hp(X,F) is a finite B-module.

Proof. To prove this we consider the fibre product diagram

X ′ = Spec(B)×Spec(A) X π//

f ′

��

X

f

��Spec(B) // Spec(A)

Note that f ′ is a proper morphism, see Morphisms of Spaces, Lemma 40.3. Also,B is a finitely generated A-algebra, and hence Noetherian (Algebra, Lemma 31.1).This implies that X ′ is a Noetherian algebraic space (Morphisms of Spaces, Lemma28.6). Note that X ′ is the relative spectrum of the quasi-coherent OX -algebra Bby Morphisms of Spaces, Lemma 20.7. Since F is a quasi-coherent B-module wesee that there is a unique quasi-coherent OX′ -module F ′ such that π∗F ′ = F ,see Morphisms of Spaces, Lemma 20.10. Since F is finite type as a B-module weconclude that F ′ is a finite type OX′ -module (details omitted). In other words, F ′is a coherent OX′ -module (Lemma 12.2). Since the morphism π : X ′ → X is affinewe have

Hp(X,F) = Hp(X ′,F ′)by Lemma 8.2 and Cohomology on Sites, Lemma 14.6. Thus the lemma followsfrom Lemma 20.3. �

21. Ample invertible sheaves and cohomology

0GF9 Here is a criterion for ampleness on proper algebraic spaces over affine bases interms of vanishing of cohomology after twisting.

Lemma 21.1.0GFA Let R be a Noetherian ring. Let X be a proper algebraic space overR. Let L be an invertible OX-module. The following are equivalent

(1) X is a scheme and L is ample on X,(2) for every coherent OX-module F there exists an n0 ≥ 0 such that Hp(X,F⊗L⊗n) = 0 for all n ≥ n0 and p > 0, and

https://stacks.math.columbia.edu/tag/08AS

https://stacks.math.columbia.edu/tag/08AT

https://stacks.math.columbia.edu/tag/0GFA


(3) for every coherent OX-module F there exists an n ≥ 1 such that H1(X,F⊗L⊗n) = 0.

Proof. The implication (1) ⇒ (2) follows from Cohomology of Schemes, Lemma17.1. The implication (2) ⇒ (3) is trivial. The implication (3) ⇒ (1) is Lemma16.9. �

Lemma 21.2.0GFB Let R be a Noetherian ring. Let f : Y → X be a morphism ofalgebraic spaces proper over R. Let L be an invertible OX-module. Assume f isfinite and surjective. The following are equivalent

(1) X is a scheme and L is ample, and(2) Y is a scheme and f∗L is ample.

Proof. Assume (1). Then Y is a scheme as a finite morphism is representable (byschemes), see Morphisms of Spaces, Lemma 45.3. Hence (2) follows from Cohomol-ogy of Schemes, Lemma 17.2.

Assume (2). Let P be the following property on coherent OX -modules F : thereexists an n0 such that Hp(X,F ⊗L⊗n) = 0 for all n ≥ n0 and p > 0. We will provethat P holds for any coherent OX -module F , which implies L is ample by Lemma21.1. We are going to apply Lemma 14.5. Thus we have to verify (1), (2) and (3) ofthat lemma for P . Property (1) follows from the long exact cohomology sequenceassociated to a short exact sequence of sheaves and the fact that tensoring withan invertible sheaf is an exact functor. Property (2) follows since Hp(X,−) is anadditive functor.

To see (3) let i : Z → X be a reduced closed subspace with |Z| irreducible. Leti′ : Z ′ → Y and f ′ : Z ′ → Z be as in Lemma 17.1 and set G = f ′∗OZ′ . We claim thatG satisfies properties (3)(a) and (3)(b) of Lemma 14.5 which will finish the proof.Property (3)(a) we have seen in Lemma 17.1. To see (3)(b) let I be a nonzeroquasi-coherent sheaf of ideals on Z. Denote I ′ ⊂ OZ′ the quasi-coherent ideal(f ′)−1IOZ′ , i.e., the image of (f ′)∗I → OZ′ . By Lemma 17.2 we have f∗I ′ = IG.We claim the common value G′ = IG = f ′∗I ′ satisfies the condition expressed in(3)(b). First, it is clear that the support of G/G′ is contained in the support ofOZ/I which is a proper subspace of |Z| as I is a nonzero ideal sheaf on the reducedand irreducible algebraic space Z. Recall that f ′∗, i∗, and i′∗ transform coherentmodules into coherent modules, see Lemmas 12.9 and 12.8. As Y is a scheme andL is ample we see from Lemma 21.1 that there exists an n0 such that

Hp(Y, i′∗I ′ ⊗OYf∗L⊗n) = 0

for n ≥ n0 and p > 0. Now we get

Hp(X, i∗G′ ⊗OXL⊗n) = Hp(Z,G′ ⊗OZ

i∗L⊗n)= Hp(Z, f ′∗I ′ ⊗OZ

i∗L⊗n))= Hp(Z, f ′∗(I ′ ⊗OZ′ (f ′)∗i∗L⊗n))= Hp(Z, f ′∗(I ′ ⊗OZ′ (i′)∗f∗L⊗n))= Hp(Z ′, I ′ ⊗OZ′ (i′)∗f∗L⊗n))= Hp(Y, i′∗I ′ ⊗OY

f∗L⊗n) = 0

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Here we have used the projection formula and the Leray spectral sequence (seeCohomology on Sites, Sections 48 and 14) and Lemma 4.1. This verifies property(3)(b) of Lemma 14.5 as desired. �

22. The theorem on formal functions

08AU This section is the analogue of Cohomology of Schemes, Section 20. We encouragethe reader to read that section first.

Situation 22.1.08AV Here A is a Noetherian ring and I ⊂ A is an ideal. Also,f : X → Spec(A) is a proper morphism of algebraic spaces and F is a coherentsheaf on X.

In this situation we denote InF the quasi-coherent submodule of F generated as anOX -module by products of local sections of F and elements of In. In other words,it is the image of the map f∗I ⊗OX

F → F .

Lemma 22.2.08AW In Situation 22.1. Set B =⊕

n≥0 In. Then for every p ≥ 0 the

graded B-module⊕

n≥0Hp(X, InF) is a finite B-module.

Proof. Let B =⊕InOX = f∗B. Then

⊕InF is a finite type graded B-module.

Hence the result follows from Lemma 20.4. �

Lemma 22.3.08AX In Situation 22.1. For every p ≥ 0 there exists an integer c ≥ 0such that

(1) the multiplication map In−c ⊗Hp(X, IcF)→ Hp(X, InF) is surjective forall n ≥ c, and

(2) the image of Hp(X, In+mF)→ Hp(X, InF) is contained in the submoduleIm−cHp(X, InF) for all n ≥ 0, m ≥ c.

Proof. By Lemma 22.2 we can find d1, . . . , dt ≥ 0, and xi ∈ Hp(X, IdiF) suchthat

⊕n≥0H

p(X, InF) is generated by x1, . . . , xt over B =⊕

n≥0 In. Take c =

max{di}. It is clear that (1) holds. For (2) let b = max(0, n − c). Consider thecommutative diagram of A-modules

In+m−c−b ⊗ Ib ⊗Hp(X, IcF) //

��

In+m−c ⊗Hp(X, IcF) // Hp(X, In+mF)

��In+m−c−b ⊗Hp(X, InF) // Hp(X, InF)

By part (1) of the lemma the composition of the horizontal arrows is surjective ifn+m ≥ c. On the other hand, it is clear that n+m− c− b ≥ m− c. Hence part(2). �

Lemma 22.4.08AY In Situation 22.1. Fix p ≥ 0.(1) There exists a c1 ≥ 0 such that for all n ≥ c1 we have

Ker(Hp(X,F)→ Hp(X,F/InF)) ⊂ In−c1Hp(X,F).

(2) The inverse system

(Hp(X,F/InF))n∈N

satisfies the Mittag-Leffler condition (see Homology, Definition 31.2).

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https://stacks.math.columbia.edu/tag/08AY


(3) In fact for any p and n there exists a c2(n) ≥ n such that

Im(Hp(X,F/IkF)→ Hp(X,F/InF)) = Im(Hp(X,F)→ Hp(X,F/InF))for all k ≥ c2(n).

Proof. Let c1 = max{cp, cp+1}, where cp, cp+1 are the integers found in Lemma22.3 for Hp and Hp+1. We will use this constant in the proofs of (1), (2) and (3).Let us prove part (1). Consider the short exact sequence

0→ InF → F → F/InF → 0From the long exact cohomology sequence we see that

Ker(Hp(X,F)→ Hp(X,F/InF)) = Im(Hp(X, InF)→ Hp(X,F))Hence by our choice of c1 we see that this is contained in In−c1Hp(X,F) for n ≥ c1.Note that part (3) implies part (2) by definition of the Mittag-Leffler condition.Let us prove part (3). Fix an n throughout the rest of the proof. Consider thecommutative diagram

0 // InF // F // F/InF // 0

0 // In+mF //

OO

F //

OO

F/In+mF //

OO

0

This gives rise to the following commutative diagram

Hp(X, InF) // Hp(X,F) // Hp(X,F/InF)δ

// Hp+1(X, InF)

Hp(X, In+mF) //

OO

Hp(X,F) //

1

OO

Hp(X,F/In+mF) //

OO

Hp+1(X, In+mF)

a

OO

If m ≥ c1 we see that the image of a is contained in Im−c1Hp+1(X, InF). By theArtin-Rees lemma (see Algebra, Lemma 51.3) there exists an integer c3(n) suchthat

INHp+1(X, InF) ∩ Im(δ) ⊂ δ(IN−c3(n)Hp(X,F/InF)

)for all N ≥ c3(n). As Hp(X,F/InF) is annihilated by In, we see that if m ≥c3(n) + c1 + n, then

Im(Hp(X,F/In+mF)→ Hp(X,F/InF)) = Im(Hp(X,F)→ Hp(X,F/InF))In other words, part (3) holds with c2(n) = c3(n) + c1 + n. �

Theorem 22.5 (Theorem on formal functions).08AZ In Situation 22.1. Fix p ≥ 0.The system of maps

Hp(X,F)/InHp(X,F) −→ Hp(X,F/InF)define an isomorphism of limits

Hp(X,F)∧ −→ limnHp(X,F/InF)

where the left hand side is the completion of the A-module Hp(X,F) with respectto the ideal I, see Algebra, Section 96. Moreover, this is in fact a homeomorphismfor the limit topologies.

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Proof. In fact, this follows immediately from Lemma 22.4. We spell out the details.Set M = Hp(X,F) and Mn = Hp(X,F/InF). Denote Nn = Im(M → Mn). Bythe description of the limit in Homology, Section 31 we have

limnMn = {(xn) ∈∏

Mn | ϕi(xn) = xn−1, n = 2, 3, . . .}

Pick an element x = (xn) ∈ limnMn. By Lemma 22.4 part (3) we have xn ∈ Nnfor all n since by definition xn is the image of some xn+m ∈ Mn+m for all m. ByLemma 22.4 part (1) we see that there exists a factorization

M → Nn →M/In−c1M

of the reduction map. Denote yn ∈ M/In−c1M the image of xn for n ≥ c1. Sincefor n′ ≥ n the composition M → Mn′ → Mn is the given map M → Mn we seethat yn′ maps to yn under the canonical map M/In

′−c1M → M/In−c1M . Hencey = (yn+c1) defines an element of limnM/InM . We omit the verification that ymaps to x under the map

M∧ = limnM/InM −→ limnMn

of the lemma. We also omit the verification on topologies. �

Lemma 22.6.08B0 Let A be a ring. Let I ⊂ A be an ideal. Assume A is Noetherianand complete with respect to I. Let f : X → Spec(A) be a proper morphism ofalgebraic spaces. Let F be a coherent sheaf on X. Then

Hp(X,F) = limnHp(X,F/InF)

for all p ≥ 0.

Proof. This is a reformulation of the theorem on formal functions (Theorem 22.5)in the case of a complete Noetherian base ring. Namely, in this case the A-moduleHp(X,F) is finite (Lemma 20.3) hence I-adically complete (Algebra, Lemma 97.1)and we see that completion on the left hand side is not necessary. �

Lemma 22.7.08B1 Let S be a scheme. Let f : X → Y be a morphism of algebraicspaces over S and let F be a quasi-coherent sheaf on Y . Assume

(1) Y locally Noetherian,(2) f proper, and(3) F coherent.

Let y be a geometric point of Y . Consider the “infinitesimal neighbourhoods”

Xn = Spec(OY,y/mny )×Y Xin//

fn

��

X

f

��Spec(OY,y/mny ) cn // Y

of the fibre X1 = Xy and set Fn = i∗nF . Then we have

(Rpf∗F)∧y ∼= limnHp(Xn,Fn)

as O∧Y,y-modules.

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Proof. This is just a reformulation of a special case of the theorem on formalfunctions, Theorem 22.5. Let us spell it out. Note that OY,y is a Noetherianlocal ring, see Properties of Spaces, Lemma 24.4. Consider the canonical morphismc : Spec(OY,y)→ Y . This is a flat morphism as it identifies local rings. Denote f ′ :X ′ → Spec(OY,y) the base change of f to this local ring. We see that c∗Rpf∗F =Rpf ′∗F ′ by Lemma 11.2. Moreover, we have canonical identifications Xn = X ′n forall n ≥ 1.Hence we may assume that Y = Spec(A) is the spectrum of a strictly henselian Noe-therian local ring A with maximal ideal m and that y → Y is equal to Spec(A/m)→Y . It follows that

(Rpf∗F)y = Γ(Y,Rpf∗F) = Hp(X,F)

because (Y, y) is an initial object in the category of étale neighbourhoods of y. Themorphisms cn are each closed immersions. Hence their base changes in are closedimmersions as well. Note that in,∗Fn = in,∗i

∗nF = F/mnF . By the Leray spectral

sequence for in, and Lemma 12.9 we see thatHp(Xn,Fn) = Hp(X, in,∗F) = Hp(X,F/mnF)

Hence we may indeed apply the theorem on formal functions to compute the limitin the statement of the lemma and we win. �

Here is a lemma which we will generalize later to fibres of dimension > 0, namelythe next lemma.

Lemma 22.8.0A4S Let S be a scheme. Let f : X → Y be a morphism of algebraicspaces over S. Let y be a geometric point of Y . Assume

(1) Y locally Noetherian,(2) f is proper, and(3) Xy has discrete underlying topological space.

Then for any coherent sheaf F on X we have (Rpf∗F)y = 0 for all p > 0.

Proof. Let κ(y) be the residue field of the local ring of OY,y. As in Lemma 22.7we set Xy = X1 = Spec(κ(y)) ×Y X. By Morphisms of Spaces, Lemma 34.8 themorphism f : X → Y is quasi-finite at each of the points of the fibre of X → Yover y. It follows that Xy → y is separated and quasi-finite. Hence Xy is a schemeby Morphisms of Spaces, Proposition 50.2. Since it is quasi-compact its underlyingtopological space is a finite discrete space. Then it is an affine scheme by Schemes,Lemma 11.8. By Lemma 17.3 it follows that the algebraic spaces Xn are affineschemes as well. Moreover, the underlying topological of each Xn is the same asthat of X1. Hence it follows that Hp(Xn,Fn) = 0 for all p > 0. Hence we see that(Rpf∗F)∧y = 0 by Lemma 22.7. Note that Rpf∗F is coherent by Lemma 20.2 andhence Rpf∗Fy is a finite OY,y-module. By Algebra, Lemma 97.1 this implies that(Rpf∗F)y = 0. �

Lemma 22.9.0A4T Let S be a scheme. Let f : X → Y be a morphism of algebraicspaces over S. Let y be a geometric point of Y . Assume

(1) Y locally Noetherian,(2) f is proper, and(3) dim(Xy) = d.

Then for any coherent sheaf F on X we have (Rpf∗F)y = 0 for all p > d.

https://stacks.math.columbia.edu/tag/0A4S

https://stacks.math.columbia.edu/tag/0A4T


Proof. Let κ(y) be the residue field of the local ring of OY,y. As in Lemma 22.7 wesetXy = X1 = Spec(κ(y))×YX. Moreover, the underlying topological space of eachinfinitesimal neighbourhood Xn is the same as that of Xy. Hence Hp(Xn,Fn) = 0for all p > d by Lemma 10.1. Hence we see that (Rpf∗F)∧y = 0 by Lemma 22.7 forp > d. Note that Rpf∗F is coherent by Lemma 20.2 and hence Rpf∗Fy is a finiteOY,y-module. By Algebra, Lemma 97.1 this implies that (Rpf∗F)y = 0. �

23. Applications of the theorem on formal functions

0A4U We will add more here as needed.

Lemma 23.1.0A4V (For a more general version see More on Morphisms of Spaces,Lemma 35.1). Let S be a scheme. Let f : X → Y be a morphism of algebraicspaces over S. Assume Y is locally Noetherian. The following are equivalent

(1) f is finite, and(2) f is proper and |Xk| is a discrete space for every morphism Spec(k) → Y

where k is a field.

Proof. A finite morphism is proper according to Morphisms of Spaces, Lemma45.9. A finite morphism is quasi-finite according to Morphisms of Spaces, Lemma45.8. A quasi-finite morphism has discrete fibres Xk, see Morphisms of Spaces,Lemma 27.5. Hence a finite morphism is proper and has discrete fibres Xk.Assume f is proper with discrete fibres Xk. We want to show f is finite. In fact itsuffices to prove f is affine. Namely, if f is affine, then it follows that f is integral byMorphisms of Spaces, Lemma 45.7 whereupon it follows from Morphisms of Spaces,Lemma 45.6 that f is finite.To show that f is affine we may assume that Y is affine, and our goal is to show thatX is affine too. Since f is proper we see that X is separated and quasi-compact.We will show that for any coherent OX -module F we have H1(X,F) = 0. Thisimplies that H1(X,F) = 0 for every quasi-coherent OX -module F by Lemmas 15.1and 5.1. Then it follows that X is affine from Proposition 16.7. By Lemma 22.8we conclude that the stalks of R1f∗F are zero for all geometric points of Y . Inother words, R1f∗F = 0. Hence we see from the Leray Spectral Sequence for f thatH1(X,F) = H1(Y, f∗F). Since Y is affine, and f∗F is quasi-coherent (Morphismsof Spaces, Lemma 11.2) we conclude H1(Y, f∗F) = 0 from Cohomology of Schemes,Lemma 2.2. Hence H1(X,F) = 0 as desired. �

As a consequence we have the following useful result.

Lemma 23.2.0A4W (For a more general version see More on Morphisms of Spaces,Lemma 35.2). Let S be a scheme. Let f : X → Y be a morphism of algebraicspaces over S. Let y be a geometric point of Y . Assume

(1) Y is locally Noetherian,(2) f is proper, and(3) |Xy| is finite.

Then there exists an open neighbourhood V ⊂ Y of y such that f |f−1(V ) : f−1(V )→V is finite.

Proof. The morphism f is quasi-finite at all the geometric points of X lying overy by Morphisms of Spaces, Lemma 34.8. By Morphisms of Spaces, Lemma 34.7 theset of points at which f is quasi-finite is an open subspace U ⊂ X. Let Z = X \U .

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Then y 6∈ f(Z). Since f is proper the set f(Z) ⊂ Y is closed. Choose any openneighbourhood V ⊂ Y of y with Z ∩ V = ∅. Then f−1(V ) → V is locally quasi-finite and proper. Hence f−1(V )→ V has discrete fibres Xk (Morphisms of Spaces,Lemma 27.5) which are quasi-compact hence finite. Thus f−1(V )→ V is finite byLemma 23.1. �

24. Other chapters

Preliminaries(1) Introduction(2) Conventions(3) Set Theory(4) Categories(5) Topology(6) Sheaves on Spaces(7) Sites and Sheaves(8) Stacks(9) Fields(10) Commutative Algebra(11) Brauer Groups(12) Homological Algebra(13) Derived Categories(14) Simplicial Methods(15) More on Algebra(16) Smoothing Ring Maps(17) Sheaves of Modules(18) Modules on Sites(19) Injectives(20) Cohomology of Sheaves(21) Cohomology on Sites(22) Differential Graded Algebra(23) Divided Power Algebra(24) Differential Graded Sheaves(25) Hypercoverings

Schemes(26) Schemes(27) Constructions of Schemes(28) Properties of Schemes(29) Morphisms of Schemes(30) Cohomology of Schemes(31) Divisors(32) Limits of Schemes(33) Varieties(34) Topologies on Schemes(35) Descent(36) Derived Categories of Schemes(37) More on Morphisms(38) More on Flatness

(39) Groupoid Schemes(40) More on Groupoid Schemes(41) Étale Morphisms of Schemes

Topics in Scheme Theory(42) Chow Homology(43) Intersection Theory(44) Picard Schemes of Curves(45) Weil Cohomology Theories(46) Adequate Modules(47) Dualizing Complexes(48) Duality for Schemes(49) Discriminants and Differents(50) de Rham Cohomology(51) Local Cohomology(52) Algebraic and Formal Geometry(53) Algebraic Curves(54) Resolution of Surfaces(55) Semistable Reduction(56) Functors and Morphisms(57) Derived Categories of Varieties(58) Fundamental Groups of Schemes(59) Étale Cohomology(60) Crystalline Cohomology(61) Pro-étale Cohomology(62) More Étale Cohomology(63) The Trace Formula

Algebraic Spaces(64) Algebraic Spaces(65) Properties of Algebraic Spaces(66) Morphisms of Algebraic Spaces(67) Decent Algebraic Spaces(68) Cohomology of Algebraic Spaces(69) Limits of Algebraic Spaces(70) Divisors on Algebraic Spaces(71) Algebraic Spaces over Fields(72) Topologies on Algebraic Spaces(73) Descent and Algebraic Spaces(74) Derived Categories of Spaces(75) More on Morphisms of Spaces(76) Flatness on Algebraic Spaces


(77) Groupoids in Algebraic Spaces(78) More on Groupoids in Spaces(79) Bootstrap(80) Pushouts of Algebraic Spaces

Topics in Geometry(81) Chow Groups of Spaces(82) Quotients of Groupoids(83) More on Cohomology of Spaces(84) Simplicial Spaces(85) Duality for Spaces(86) Formal Algebraic Spaces(87) Algebraization of Formal Spaces(88) Resolution of Surfaces Revisited

Deformation Theory(89) Formal Deformation Theory(90) Deformation Theory(91) The Cotangent Complex(92) Deformation Problems

Algebraic Stacks(93) Algebraic Stacks(94) Examples of Stacks(95) Sheaves on Algebraic Stacks(96) Criteria for Representability

(97) Artin’s Axioms(98) Quot and Hilbert Spaces(99) Properties of Algebraic Stacks(100) Morphisms of Algebraic Stacks(101) Limits of Algebraic Stacks(102) Cohomology of Algebraic Stacks(103) Derived Categories of Stacks(104) Introducing Algebraic Stacks(105) More on Morphisms of Stacks(106) The Geometry of Stacks

Topics in Moduli Theory(107) Moduli Stacks(108) Moduli of Curves

Miscellany(109) Examples(110) Exercises(111) Guide to Literature(112) Desirables(113) Coding Style(114) Obsolete(115) GNU Free Documentation Li-

cense(116) Auto Generated Index

References[Knu71] Donald Knutson, Algebraic spaces, Lecture Notes in Mathematics, vol. 203, Springer-

Verlag, 1971.

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